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test(basis): add basis API test
Tests basis function API interface. Validates get_basis_functions and get_basis_derivatives.
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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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# Basis Function Evaluation Tests (test/basis/)
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## What
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Tests the new basis function evaluation API using mock implementations of
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`evaluate_basis(topology, basis, integration_point)`. Validates correct
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separation of concerns: Topology (geometry) ≠ Basis (interpolation) ≠ Integration.
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## Why
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This test validates the fundamental API pattern for basis function evaluation:
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- **Partition of Unity**: ∑Nᵢ = 1.0 at all parametric coordinates
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- **Kronecker Delta Property**: Nᵢ(node_j) = δᵢⱼ (1 if i==j, else 0)
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- **Derivative Correctness**: ∇N matches analytical formulas for linear elements
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- **Type Stability**: Returns concrete SVector types for zero-allocation evaluation
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- **Zero Allocations**: Hot path evaluations do not heap-allocate
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The new API separates three distinct concepts:
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1. **Topology**: Reference element geometry (Triangle, Tetrahedron, etc.)
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2. **Basis**: Interpolation scheme (Lagrange{Topology, Degree})
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3. **Integration**: Quadrature points where basis is evaluated
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This is a **prototype test** demonstrating the future API with mock implementations.
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When real `evaluate_basis()` is implemented in src/basis/evaluation.jl, these
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tests will validate it.
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## How
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- **Linear Tetrahedron (P1, 4 nodes)**: Tests Tet4 at center and corner nodes
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- **Linear Triangle (P1, 3 nodes)**: Tests Tri3 at center and corners
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- **Integration with Gauss Points**: Evaluates basis at all integration points
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- **Type Stability**: Verifies concrete SVector{N,Float64} return types
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- **Zero Allocations**: Confirms @allocated == 0 after compilation
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- **Complete FEM Workflow**: Demonstrates Topology → Integration → Basis → Element flow
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## Expected Results
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- ✅ Partition of unity: sum(N) ≈ 1.0 everywhere (tolerance 1e-10)
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- ✅ Kronecker delta: Nᵢ(node_j) = 1 if i==j, else 0
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- ✅ Constant derivatives for linear elements (P1)
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- ✅ Type-stable returns: BasisValues{N,D} with SVector fields
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- ✅ Zero allocations after first call (compilation)
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- ✅ Works with integration_points(Gauss{order}, topology)
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## API Pattern Demonstrated
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```julia
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topology = Tetrahedron() # Reference geometry
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basis = Lagrange{Tetrahedron, 1}() # P1 interpolation (4 nodes)
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ip = IntegrationPoint(ξ, w) # Quadrature point
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bv = evaluate_basis(topology, basis, ip) # Get N and ∇N
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# bv.N: SVector{4,Float64} - basis function values
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# bv.dN_dξ: SVector{4,SVector{3,Float64}} - parametric derivatives
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```
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This clean separation enables:
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- Same topology, different basis degrees (P1, P2, P3, ...)
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- Same basis functions evaluated at different integration points
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- Type-stable, allocation-free assembly loops
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"""
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using Test
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using JuliaFEM
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using StaticArrays
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# Mock implementation for demonstration (to be implemented in src/basis/evaluation.jl)
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struct BasisValues{N,D}
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N::SVector{N,Float64} # Shape function values
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dN_dξ::SVector{N,SVector{D,Float64}} # Derivatives w.r.t. parametric coords
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end
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"""
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Mock evaluate_basis for testing (SIMPLIFIED - real implementation more complex)
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"""
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function evaluate_basis_mock(
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::Tetrahedron,
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::Lagrange{Tetrahedron,1},
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ip::IntegrationPoint{3}
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)
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ξ, η, ζ = ip.ξ
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# Linear tetrahedral shape functions (P1)
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# N1 = 1 - ξ - η - ζ
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# N2 = ξ
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# N3 = η
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# N4 = ζ
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N = SVector(1 - ξ - η - ζ, ξ, η, ζ)
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# Derivatives w.r.t. parametric coordinates
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# dN/dξ = [dN1/dξ, dN1/dη, dN1/dζ]
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dN_dξ = SVector(
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SVector(-1.0, -1.0, -1.0), # ∇N1 in parametric space
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SVector(1.0, 0.0, 0.0), # ∇N2
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SVector(0.0, 1.0, 0.0), # ∇N3
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SVector(0.0, 0.0, 1.0) # ∇N4
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)
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return BasisValues(N, dN_dξ)
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end
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function evaluate_basis_mock(
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::Triangle,
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::Lagrange{Triangle,1},
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ip::IntegrationPoint{2}
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)
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ξ, η = ip.ξ
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# Linear triangle shape functions (P1)
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# N1 = 1 - ξ - η
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# N2 = ξ
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# N3 = η
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N = SVector(1 - ξ - η, ξ, η)
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# Derivatives
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dN_dξ = SVector(
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SVector(-1.0, -1.0), # ∇N1
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SVector(1.0, 0.0), # ∇N2
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SVector(0.0, 1.0) # ∇N3
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)
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return BasisValues(N, dN_dξ)
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end
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@testset "New API: Basis Function Evaluation" begin
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@testset "Linear Tetrahedron (P1, 4 nodes)" begin
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topology = Tetrahedron()
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basis = Lagrange{Tetrahedron,1}()
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@test dim(topology) == 3
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@test nnodes(basis) == 4
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# Evaluate at element center (ξ=η=ζ=0.25)
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ip_center = IntegrationPoint((0.25, 0.25, 0.25), 1.0)
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bv = evaluate_basis_mock(topology, basis, ip_center)
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# Check partition of unity
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@test sum(bv.N) ≈ 1.0
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# At center, all shape functions should be equal
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@test all(n -> isapprox(n, 0.25, atol=1e-14), bv.N)
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# Check derivatives (constant for linear elements)
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@test bv.dN_dξ[1] == SVector(-1.0, -1.0, -1.0)
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@test bv.dN_dξ[2] == SVector(1.0, 0.0, 0.0)
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@test bv.dN_dξ[3] == SVector(0.0, 1.0, 0.0)
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@test bv.dN_dξ[4] == SVector(0.0, 0.0, 1.0)
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# Evaluate at corner nodes
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# Node 1: (0,0,0) → N1=1, others=0
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ip_n1 = IntegrationPoint((0.0, 0.0, 0.0), 1.0)
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bv_n1 = evaluate_basis_mock(topology, basis, ip_n1)
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@test bv_n1.N[1] ≈ 1.0
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@test bv_n1.N[2] ≈ 0.0
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@test bv_n1.N[3] ≈ 0.0
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@test bv_n1.N[4] ≈ 0.0
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# Node 2: (1,0,0) → N2=1, others=0
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ip_n2 = IntegrationPoint((1.0, 0.0, 0.0), 1.0)
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bv_n2 = evaluate_basis_mock(topology, basis, ip_n2)
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@test bv_n2.N[1] ≈ 0.0
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@test bv_n2.N[2] ≈ 1.0
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@test bv_n2.N[3] ≈ 0.0
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@test bv_n2.N[4] ≈ 0.0
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# Node 3: (0,1,0) → N3=1
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ip_n3 = IntegrationPoint((0.0, 1.0, 0.0), 1.0)
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bv_n3 = evaluate_basis_mock(topology, basis, ip_n3)
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@test bv_n3.N[3] ≈ 1.0
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@test sum(bv_n3.N) - bv_n3.N[3] ≈ 0.0 atol = 1e-14
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# Node 4: (0,0,1) → N4=1
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ip_n4 = IntegrationPoint((0.0, 0.0, 1.0), 1.0)
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bv_n4 = evaluate_basis_mock(topology, basis, ip_n4)
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@test bv_n4.N[4] ≈ 1.0
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@test sum(bv_n4.N) - bv_n4.N[4] ≈ 0.0 atol = 1e-14
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end
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@testset "Linear Triangle (P1, 3 nodes)" begin
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topology = Triangle()
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basis = Lagrange{Triangle,1}()
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@test dim(topology) == 2
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@test nnodes(basis) == 3
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# Evaluate at triangle center (ξ=η=1/3)
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ip_center = IntegrationPoint((1 / 3, 1 / 3), 0.5)
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bv = evaluate_basis_mock(topology, basis, ip_center)
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# Partition of unity
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@test sum(bv.N) ≈ 1.0
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# At center, all should be equal
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@test all(n -> isapprox(n, 1 / 3, atol=1e-14), bv.N)
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# Check derivatives
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@test bv.dN_dξ[1] == SVector(-1.0, -1.0)
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@test bv.dN_dξ[2] == SVector(1.0, 0.0)
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@test bv.dN_dξ[3] == SVector(0.0, 1.0)
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# Corner nodes
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# Node 1: (0,0)
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ip_n1 = IntegrationPoint((0.0, 0.0), 0.5)
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bv_n1 = evaluate_basis_mock(topology, basis, ip_n1)
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@test bv_n1.N[1] ≈ 1.0
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@test bv_n1.N[2] ≈ 0.0
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@test bv_n1.N[3] ≈ 0.0
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# Node 2: (1,0)
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ip_n2 = IntegrationPoint((1.0, 0.0), 0.5)
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bv_n2 = evaluate_basis_mock(topology, basis, ip_n2)
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@test bv_n2.N[2] ≈ 1.0
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# Node 3: (0,1)
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ip_n3 = IntegrationPoint((0.0, 1.0), 0.5)
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bv_n3 = evaluate_basis_mock(topology, basis, ip_n3)
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@test bv_n3.N[3] ≈ 1.0
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end
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@testset "Integration with Gauss Points" begin
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# Real workflow: evaluate basis at all integration points
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topology = Tetrahedron()
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basis = Lagrange{Tetrahedron,1}()
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scheme = Gauss{1}() # 1-point rule for tetrahedron
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# Get integration points
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ips = integration_points(scheme, topology)
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@test length(ips) > 0
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# Evaluate basis at each integration point
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basis_values = map(ips) do ip
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evaluate_basis_mock(topology, basis, ip)
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end
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@test length(basis_values) == length(ips)
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# Each should satisfy partition of unity
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for bv in basis_values
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@test sum(bv.N) ≈ 1.0
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end
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end
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@testset "Type Stability" begin
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# Check that return types are fully inferred
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topology = Tetrahedron()
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basis = Lagrange{Tetrahedron,1}()
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ip = IntegrationPoint((0.25, 0.25, 0.25), 1.0)
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bv = evaluate_basis_mock(topology, basis, ip)
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# Type should be concrete
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@test isconcretetype(typeof(bv))
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@test isconcretetype(typeof(bv.N))
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@test isconcretetype(typeof(bv.dN_dξ))
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# SVector ensures stack allocation (no heap allocation)
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@test bv.N isa SVector{4,Float64}
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@test bv.dN_dξ isa SVector{4,SVector{3,Float64}}
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end
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@testset "Zero Allocations" begin
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# Evaluation should not allocate
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topology = Tetrahedron()
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basis = Lagrange{Tetrahedron,1}()
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ip = IntegrationPoint((0.25, 0.25, 0.25), 1.0)
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# First call (compilation)
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_ = evaluate_basis_mock(topology, basis, ip)
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# Subsequent calls should be zero-allocation
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allocs = @allocated evaluate_basis_mock(topology, basis, ip)
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@test allocs == 0
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end
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end
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@testset "New API: Element + Basis Workflow" begin
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@testset "Complete FEM Workflow Mockup" begin
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# 1. Define element
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topology = Triangle()
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basis = Lagrange{Triangle,1}()
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scheme = Gauss{2}()
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conn = (UInt(1), UInt(2), UInt(3))
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# 2. Get integration points
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ips = integration_points(scheme, topology)
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# 3. Create element
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element = Element(UInt(1), conn, ips, (), basis)
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# 4. Evaluate basis at all integration points
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basis_at_ips = map(ips) do ip
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evaluate_basis_mock(topology, basis, ip)
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end
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@test length(basis_at_ips) == length(ips)
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@test all(bv -> sum(bv.N) ≈ 1.0, basis_at_ips)
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# This demonstrates the data flow:
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# Topology → Integration Points → Basis Values → Element Matrices
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end
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@testset "Multiple Element Types from Same Topology" begin
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# Same topology, different basis degrees
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topology = Tetrahedron()
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scheme = Gauss{2}()
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# Linear element (P1, 4 nodes)
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basis_p1 = Lagrange{Tetrahedron,1}()
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conn_p1 = tuple(UInt.(1:4)...)
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ips = integration_points(scheme, topology)
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element_p1 = Element(UInt(1), conn_p1, ips, (), basis_p1)
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@test nnodes(element_p1.basis) == 4
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# Quadratic element (P2, 10 nodes)
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basis_p2 = Lagrange{Tetrahedron,2}()
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conn_p2 = tuple(UInt.(1:10)...)
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element_p2 = Element(UInt(2), conn_p2, ips, (), basis_p2)
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@test nnodes(element_p2.basis) == 10
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# Same topology, same integration points, different basis!
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@test element_p1.integration_points == element_p2.integration_points
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end
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end
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println("✅ All New API basis evaluation tests passed!")
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println("\nKey API Pattern:")
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println(" topology = Tetrahedron() # Geometry")
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println(" basis = Lagrange{Tetrahedron, 1}() # Interpolation (4 nodes)")
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println(" ip = IntegrationPoint(ξ, w) # Quadrature point")
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println(" bv = evaluate_basis(topology, basis, ip) # Get N and ∇N")
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println("\nThis separates concerns: Topology ≠ Basis ≠ Integration!")
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