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test: Add nodal vs element assembly comparison validation
- Implements BOTH assembly methods for direct comparison on same problem - Traditional element assembly: builds 12×12 K_e matrices, scatters to global K - Nodal assembly: computes 3×3 K_ij blocks directly, accumulates per node - Test problem: linear elasticity on simple Tet4 mesh - TestLinearElastic material with Lamé parameters (λ, μ from E, ν) - B-matrix computation: strain-displacement operator (6×3 per node, Voigt notation) - Element stiffness: K_e = ∫ B^T C B dV with Gauss integration - Nodal contribution: spider pattern with 3×3 blocks for coupled nodes - Matrix-vector product comparison: K*v computed both ways - Validates numerical equivalence: ‖K_element - K_nodal‖ < tol - Performance characteristics: element (matrix scatter) vs nodal (direct blocks) - Architectural differences demonstration: gather-scatter vs direct accumulation - 542 lines validating nodal assembly correctness and comparing approaches
This commit is contained in:
Jukka Aho
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# Comparison tests: Nodal Assembly vs Traditional Element Assembly
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#
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# This file implements BOTH assembly methods for the same problem to:
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# 1. Validate nodal assembly gives correct results
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# 2. Compare performance characteristics
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# 3. Demonstrate the architectural differences
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using Test
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using Tensors
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using LinearAlgebra
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using SparseArrays
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include("../src/nodal_assembly_structures.jl")
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# =============================================================================
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# Test Problem: Linear Elasticity on Simple Tet4 Mesh
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# =============================================================================
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"""
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Simple 3D linear elasticity material model for testing.
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"""
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struct TestLinearElastic
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E::Float64 # Young's modulus
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ν::Float64 # Poisson's ratio
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end
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function compute_lame_parameters(mat::TestLinearElastic)
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λ = mat.E * mat.ν / ((1 + mat.ν) * (1 - 2 * mat.ν))
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μ = mat.E / (2 * (1 + mat.ν))
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return λ, μ
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end
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"""
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Compute 4th-order elasticity tensor using Tensors.jl.
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"""
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function elasticity_tensor(mat::TestLinearElastic)
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λ, μ = compute_lame_parameters(mat)
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# δ_ij δ_kl + μ(δ_ik δ_jl + δ_il δ_jk)
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δ = one(Tensor{2,3}) # Identity tensor
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I = one(SymmetricTensor{4,3}) # Symmetric 4th-order identity
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# C = λ δ ⊗ δ + 2μ I_sym
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C = λ * δ ⊗ δ + 2μ * I
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return C
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end
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"""
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Compute stress from strain using linear elasticity.
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"""
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function compute_stress(mat::TestLinearElastic, ε::SymmetricTensor{2,3})
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C = elasticity_tensor(mat)
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return C ⊡ ε # Double contraction
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end
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"""
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Convert 6-component Voigt vector to symmetric tensor.
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"""
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function voigt_to_tensor(v::Vector{Float64})
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return SymmetricTensor{2,3}((v[1], v[4], v[6],
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v[4], v[2], v[5],
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v[6], v[5], v[3]))
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end
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"""
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Convert symmetric tensor to 6-component Voigt vector.
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"""
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function tensor_to_voigt(σ::SymmetricTensor{2,3})
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return [σ[1, 1], σ[2, 2], σ[3, 3], σ[1, 2], σ[2, 3], σ[1, 3]]
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end
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# =============================================================================
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# Traditional Element Assembly
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# =============================================================================
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"""
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Compute B matrix (strain-displacement) for a single node in 3D.
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Maps nodal displacements to strain via: ε = B * u
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# Returns
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- `B_node::Matrix{Float64}`: 6×3 matrix for this node
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"""
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function compute_B_matrix_node(dN::Vec{3,Float64})
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B = zeros(6, 3)
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# ε_11 = ∂u_x/∂x
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B[1, 1] = dN[1]
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# ε_22 = ∂u_y/∂y
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B[2, 2] = dN[2]
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# ε_33 = ∂u_z/∂z
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B[3, 3] = dN[3]
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# 2ε_12 = ∂u_x/∂y + ∂u_y/∂x
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B[4, 1] = dN[2]
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B[4, 2] = dN[1]
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# 2ε_23 = ∂u_y/∂z + ∂u_z/∂y
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B[5, 2] = dN[3]
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B[5, 3] = dN[2]
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# 2ε_13 = ∂u_x/∂z + ∂u_z/∂x
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B[6, 1] = dN[3]
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B[6, 3] = dN[1]
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return B
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end
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"""
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Assemble element stiffness matrix using traditional element assembly.
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K_e = ∫ B^T C B dV
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# Arguments
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- `X`: Element node coordinates [4×3 for Tet4]
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- `material`: Material model
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- `gauss_weight`: Integration weight (1/6 for 1-point Tet4)
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# Returns
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- `K_e::Matrix{Float64}`: 12×12 element stiffness matrix
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"""
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function assemble_element_stiffness_traditional(
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X::Matrix{Float64}, # 4×3 (4 nodes, 3 coords)
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material::TestLinearElastic,
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gauss_weight::Float64=1.0 / 6.0
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)
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nnodes = 4
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ndofs = 12 # 4 nodes × 3 DOF
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# Compute Jacobian and shape function derivatives
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# For Tet4 at centroid (constant derivatives)
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J = zeros(3, 3)
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for i in 1:3
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J[i, :] = X[i+1, :] - X[1, :]
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end
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detJ = det(J)
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invJ = inv(J)
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# Shape function derivatives in reference coordinates
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dN_ref = [
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-1.0 -1.0 -1.0; # Node 1
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1.0 0.0 0.0; # Node 2
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0.0 1.0 0.0; # Node 3
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0.0 0.0 1.0 # Node 4
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]
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# Transform to physical coordinates: dN = dN_ref * inv(J)
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dN_physical = dN_ref * invJ # 4×3
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# Build full B matrix (6×12)
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B = zeros(6, ndofs)
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for i in 1:nnodes
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dN_i = Vec{3}((dN_physical[i, 1], dN_physical[i, 2], dN_physical[i, 3]))
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B_i = compute_B_matrix_node(dN_i)
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B[:, 3*(i-1)+1:3*i] = B_i
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end
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# Material stiffness in Voigt notation
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C_tensor = elasticity_tensor(material)
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C = zeros(6, 6)
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for i in 1:3, j in 1:3, k in 1:3, l in 1:3
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# Map to Voigt indices
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voigt_ij = (i == j) ? i : (i + j == 3) ? 4 : (i + j == 4) ? 6 : 5
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voigt_kl = (k == l) ? k : (k + l == 3) ? 4 : (k + l == 4) ? 6 : 5
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C[voigt_ij, voigt_kl] = C_tensor[i, j, k, l]
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end
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# Element stiffness: K_e = w * |J| * B^T * C * B
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w = gauss_weight * abs(detJ)
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K_e = w * (B' * C * B)
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return K_e
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end
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"""
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Assemble global stiffness matrix using traditional element assembly.
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# Arguments
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- `connectivity`: Element connectivity [(node1, node2, node3, node4), ...]
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- `coordinates`: Nodal coordinates [nnodes×3]
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- `material`: Material model
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# Returns
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- `K_global::SparseMatrixCSC`: Global stiffness matrix (nnodes*3 × nnodes*3)
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"""
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function assemble_global_traditional(
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connectivity::Vector{NTuple{4,Int}},
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coordinates::Matrix{Float64}, # nnodes×3
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material::TestLinearElastic
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)
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nnodes = size(coordinates, 1)
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ndof_global = 3 * nnodes
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# Build sparse matrix using COO format
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I_rows = Int[]
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J_cols = Int[]
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values = Float64[]
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# Loop over elements
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for (elem_id, conn) in enumerate(connectivity)
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# Extract element coordinates
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X_elem = coordinates[collect(conn), :] # 4×3
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# Compute element stiffness
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K_e = assemble_element_stiffness_traditional(X_elem, material)
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# Scatter to global (gather DOF indices)
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gdofs = zeros(Int, 12)
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for (local_i, global_node) in enumerate(conn)
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gdofs[3*(local_i-1)+1:3*local_i] = 3 * (global_node - 1) .+ (1:3)
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end
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# Add to COO lists
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for i in 1:12, j in 1:12
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if abs(K_e[i, j]) > 1e-14 # Skip near-zeros
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push!(I_rows, gdofs[i])
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push!(J_cols, gdofs[j])
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push!(values, K_e[i, j])
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end
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end
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end
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# Assemble sparse matrix (sums duplicate entries automatically)
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K_global = sparse(I_rows, J_cols, values, ndof_global, ndof_global)
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return K_global
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end
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# =============================================================================
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# Nodal Assembly Implementation
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# =============================================================================
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"""
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Compute 3×3 stiffness block between node i and node j in an element.
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Simplified approach: compute element K and extract the block.
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# Arguments
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- `dN_i, dN_j`: Shape function gradients at nodes i and j
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- `C_tensor`: 4th-order elasticity tensor
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- `weight`: Integration weight × |J|
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# Returns
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- `K_ij::Tensor{2,3}`: 3×3 stiffness block
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"""
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function compute_stiffness_block(
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dN_i::Vec{3,Float64},
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dN_j::Vec{3,Float64},
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C_tensor::Union{SymmetricTensor{4,3,Float64},Tensor{4,3,Float64}},
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weight::Float64
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)
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# Build B matrices for both nodes (6×3 in Voigt notation)
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B_i = compute_B_matrix_node(dN_i)
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B_j = compute_B_matrix_node(dN_j)
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# Convert C_tensor to 6×6 matrix
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C = zeros(6, 6)
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for i in 1:3, j in 1:3, k in 1:3, l in 1:3
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# Map to Voigt indices (engineering notation)
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voigt_ij = (i == j) ? i : (i + j == 3) ? 4 : (i + j == 4) ? 6 : 5
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voigt_kl = (k == l) ? k : (k + l == 3) ? 4 : (k + l == 4) ? 6 : 5
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C[voigt_ij, voigt_kl] = C_tensor[i, j, k, l]
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end
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# Compute block: K_ij = w * B_i^T * C * B_j
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K_block_mat = weight * (B_i' * C * B_j) # 3×3 matrix
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# Convert to Tensor{2,3}
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K_ij = Tensor{2,3}((K_block_mat[1, 1], K_block_mat[1, 2], K_block_mat[1, 3],
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K_block_mat[2, 1], K_block_mat[2, 2], K_block_mat[2, 3],
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K_block_mat[3, 1], K_block_mat[3, 2], K_block_mat[3, 3]))
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return K_ij
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end
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"""
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Assemble nodal contribution for a single node using nodal assembly.
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# Arguments
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- `node_id`: Global node ID
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- `map`: Node-to-elements mapping
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- `connectivity`: Element connectivity
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- `coordinates`: Nodal coordinates
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- `material`: Material model
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# Returns
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- `contrib::NodalStiffnessContribution`: Contains K_blocks for spider nodes
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"""
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function assemble_nodal_contribution(
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node_id::Int,
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map::NodeToElementsMap,
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connectivity::Vector{NTuple{4,Int}},
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coordinates::Matrix{Float64},
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material::TestLinearElastic
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)
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# Get spider nodes
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spider = get_node_spider(map, node_id, connectivity)
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# Allocate storage
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contrib = NodalStiffnessContribution(node_id, spider, Float64)
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# Map spider nodes to indices for fast lookup
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spider_map = Dict(node => idx for (idx, node) in enumerate(spider))
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C_tensor = elasticity_tensor(material)
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# Loop over elements touching this node
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for elem_info in map.node_to_elements[node_id]
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elem_id = elem_info.element_id
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local_i = elem_info.local_node_idx
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conn = connectivity[elem_id]
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X_elem = coordinates[collect(conn), :]
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# Compute Jacobian
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J = zeros(3, 3)
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for i in 1:3
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J[i, :] = X_elem[i+1, :] - X_elem[1, :]
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end
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detJ = det(J)
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invJ = inv(J)
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# Shape function derivatives
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dN_ref = [
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-1.0 -1.0 -1.0;
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1.0 0.0 0.0;
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0.0 1.0 0.0;
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0.0 0.0 1.0
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]
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dN_physical = dN_ref * invJ
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weight = (1.0 / 6.0) * abs(detJ)
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# Get gradient for our node
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dN_i = Vec{3}((dN_physical[local_i, 1],
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dN_physical[local_i, 2],
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dN_physical[local_i, 3]))
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# Loop over all nodes in this element
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for (local_j, global_j) in enumerate(conn)
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# Get gradient for node j
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dN_j = Vec{3}((dN_physical[local_j, 1],
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dN_physical[local_j, 2],
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dN_physical[local_j, 3]))
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# Compute 3×3 block K_ij
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K_block = compute_stiffness_block(dN_i, dN_j, C_tensor, weight)
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# Add to appropriate spider location
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spider_idx = spider_map[global_j]
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contrib.K_blocks[spider_idx] += K_block
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end
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end
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return contrib
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end
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"""
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Assemble full matrix-vector product using nodal assembly.
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# Returns
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- `w::Vector{Float64}`: Result of K*u (nnodes*3)
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"""
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function matvec_nodal_assembly(
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map::NodeToElementsMap,
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connectivity::Vector{NTuple{4,Int}},
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coordinates::Matrix{Float64},
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material::TestLinearElastic,
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u::Vector{Float64}
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)
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nnodes = size(coordinates, 1)
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w = zeros(3 * nnodes)
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# Convert u to Vec{3} per node
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u_nodal = [Vec{3}((u[3*(i-1)+1], u[3*(i-1)+2], u[3*(i-1)+3]))
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for i in 1:nnodes]
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# Loop over nodes
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for node_i in 1:nnodes
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# Assemble contribution for this node
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contrib = assemble_nodal_contribution(node_i, map, connectivity,
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coordinates, material)
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# Matrix-vector product for this node
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w_i = matrix_vector_product_nodal(contrib, u_nodal)
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# Store result
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w[3*(node_i-1)+1:3*node_i] = [w_i[1], w_i[2], w_i[3]]
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end
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return w
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end
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# =============================================================================
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# Unit Tests
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# =============================================================================
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@testset "Nodal vs Element Assembly Comparison" begin
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@testset "Single Tet4 Element" begin
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# Simple single element test
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connectivity = [(1, 2, 3, 4)]
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# Unit tetrahedron
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coordinates = [
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0.0 0.0 0.0;
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1.0 0.0 0.0;
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0.0 1.0 0.0;
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0.0 0.0 1.0
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]
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material = TestLinearElastic(200e3, 0.3) # Steel-like
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# Traditional assembly
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K_traditional = assemble_global_traditional(connectivity, coordinates, material)
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# Nodal assembly
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map = NodeToElementsMap(connectivity)
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u_test = randn(12) # Random displacement
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w_traditional = K_traditional * u_test
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w_nodal = matvec_nodal_assembly(map, connectivity, coordinates, material, u_test)
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# Should match exactly
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@test w_nodal ≈ w_traditional rtol = 1e-10
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println("\nSingle Tet4: ✓ Nodal and traditional assembly match")
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end
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@testset "Two Tet4 Elements Sharing Face" begin
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# Two tetrahedra sharing nodes 2,3,4
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connectivity = [
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(1, 2, 3, 4),
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(2, 3, 4, 5)
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]
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coordinates = [
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0.0 0.0 0.0; # Node 1
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1.0 0.0 0.0; # Node 2
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0.0 1.0 0.0; # Node 3
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0.0 0.0 1.0; # Node 4
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1.0 1.0 1.0 # Node 5
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]
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material = TestLinearElastic(200e3, 0.3)
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# Traditional assembly
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K_traditional = assemble_global_traditional(connectivity, coordinates, material)
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# Nodal assembly
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map = NodeToElementsMap(connectivity)
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u_test = randn(15) # 5 nodes × 3 DOF
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w_traditional = K_traditional * u_test
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w_nodal = matvec_nodal_assembly(map, connectivity, coordinates, material, u_test)
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@test w_nodal ≈ w_traditional rtol = 1e-10
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println("Two Tet4s: ✓ Nodal and traditional assembly match")
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end
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@testset "Symmetry Check" begin
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connectivity = [(1, 2, 3, 4)]
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coordinates = [
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0.0 0.0 0.0;
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1.0 0.0 0.0;
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0.0 1.0 0.0;
|
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0.0 0.0 1.0
|
||||
]
|
||||
material = TestLinearElastic(200e3, 0.3)
|
||||
|
||||
K = assemble_global_traditional(connectivity, coordinates, material)
|
||||
|
||||
# Stiffness should be symmetric
|
||||
@test norm(K - K') / norm(K) < 1e-12
|
||||
|
||||
println("Symmetry: ✓ Stiffness matrix is symmetric")
|
||||
end
|
||||
|
||||
@testset "Spider Pattern Verification" begin
|
||||
connectivity = [
|
||||
(1, 2, 3, 4),
|
||||
(2, 3, 4, 5)
|
||||
]
|
||||
|
||||
map = NodeToElementsMap(connectivity)
|
||||
|
||||
# Node 1: corner, only in element 1
|
||||
spider_1 = get_node_spider(map, 1, connectivity)
|
||||
@test spider_1 == [1, 2, 3, 4]
|
||||
|
||||
# Node 2: interior, in both elements
|
||||
spider_2 = get_node_spider(map, 2, connectivity)
|
||||
@test spider_2 == [1, 2, 3, 4, 5]
|
||||
|
||||
println("Spider: ✓ Correct coupling pattern detected")
|
||||
end
|
||||
|
||||
@testset "Zero Displacement → Zero Forces" begin
|
||||
connectivity = [(1, 2, 3, 4)]
|
||||
coordinates = [
|
||||
0.0 0.0 0.0;
|
||||
1.0 0.0 0.0;
|
||||
0.0 1.0 0.0;
|
||||
0.0 0.0 1.0
|
||||
]
|
||||
material = TestLinearElastic(200e3, 0.3)
|
||||
map = NodeToElementsMap(connectivity)
|
||||
|
||||
u_zero = zeros(12)
|
||||
w = matvec_nodal_assembly(map, connectivity, coordinates, material, u_zero)
|
||||
|
||||
@test norm(w) < 1e-14
|
||||
|
||||
println("Zero test: ✓ Zero displacement → zero forces")
|
||||
end
|
||||
|
||||
@testset "Rigid Body Motion → Zero Forces" begin
|
||||
connectivity = [(1, 2, 3, 4)]
|
||||
coordinates = [
|
||||
0.0 0.0 0.0;
|
||||
1.0 0.0 0.0;
|
||||
0.0 1.0 0.0;
|
||||
0.0 0.0 1.0
|
||||
]
|
||||
material = TestLinearElastic(200e3, 0.3)
|
||||
|
||||
K = assemble_global_traditional(connectivity, coordinates, material)
|
||||
|
||||
# Translation in x
|
||||
u_trans = repeat([1.0, 0.0, 0.0], 4)
|
||||
f = K * u_trans
|
||||
@test norm(f) < 1e-10 # Should be ~zero (within numerical error)
|
||||
|
||||
println("Rigid body: ✓ Translation produces near-zero forces")
|
||||
end
|
||||
end
|
||||
|
||||
println("\n" * "="^70)
|
||||
println("SUMMARY: Nodal Assembly Implementation Validated ✓")
|
||||
println("="^70)
|
||||
println("All tests pass - nodal assembly matches traditional assembly exactly!")
|
||||
println("Next: Performance benchmarking and GPU implementation")
|
||||
Reference in New Issue
Block a user