Step 7: local-stiffness-kernel

Read These Files First

Read the following files before editing:

/AGENTS.md
/docs/ARCHITECTURE.md
/docs/ADR.md
/docs/formulations/euler-beam-3d-formulation.md
/docs/numerical-reviews/euler-beam-3d-review.md
/docs/implementation-plans/euler-beam-3d-implementation-plan.md

Also read any files created by previous steps in this phase.

Task

Use TDD to create the local 12x12 stiffness kernel for the 3D Euler-Bernoulli beam.

Create:

/src/fesa/elements/euler_beam_3d.hpp
/src/fesa/elements/euler_beam_3d.cpp
/tests/unit/euler_beam_3d_local_stiffness_test.cpp

Required API:

namespace fesa::elements {

using Vector12 = std::array<double, 12>;
using Matrix12 = std::array<double, 144>;

struct EulerBeam3DSection {
    double young_modulus;
    double shear_modulus;
    double area;
    double torsion_constant;
    double second_moment_y;
    double second_moment_z;
};

Matrix12 euler_beam_3d_local_stiffness(double length, const EulerBeam3DSection& section);
Vector12 euler_beam_3d_local_end_forces(double length,
                                        const EulerBeam3DSection& section,
                                        const Vector12& local_displacements);

} // namespace fesa::elements

Implementation rules:

Matrix storage is row-major: index (row, col) is row * 12 + col.
Validate length > 0 and all section constants are positive; throw std::invalid_argument otherwise.
Use only C++17 standard library.
Do not introduce an external linear algebra dependency.
Do not edit CMake files; source and test globs should pick up new files.

Tests To Write First

Create /tests/unit/euler_beam_3d_local_stiffness_test.cpp before production code.

Test behavior:

For L = 2.0, E = 210.0, G = 80.0, A = 3.0, J = 4.0, Iy = 5.0, Iz = 6.0, verify representative matrix entries:
- axial: K(0,0) = EA/L, K(0,6) = -EA/L, K(6,6) = EA/L
- torsion: K(3,3) = GJ/L, K(3,9) = -GJ/L, K(9,9) = GJ/L
- local x-y bending uses EIz: K(1,1) = 12*E*Iz/L^3, K(1,5) = 6*E*Iz/L^2, K(5,5) = 4*E*Iz/L
- local x-z bending uses EIy: K(2,2) = 12*E*Iy/L^3, K(2,4) = -6*E*Iy/L^2, K(4,4) = 4*E*Iy/L
Verify all entries are symmetric within 1.0e-10.
Verify euler_beam_3d_local_end_forces returns K * u for a displacement vector with at least three nonzero components.
Verify invalid length and nonpositive section constants throw std::invalid_argument.

RED command:

ctest --test-dir build/msvc-debug --output-on-failure -C Debug -R euler_beam_3d_local_stiffness_test

Expected RED: test executable missing or compile failure because the header/API does not exist.

Then implement the minimal API.

GREEN command:

ctest --test-dir build/msvc-debug --output-on-failure -C Debug -R euler_beam_3d_local_stiffness_test

Acceptance Criteria

ctest --test-dir build/msvc-debug --output-on-failure -C Debug -R euler_beam_3d_local_stiffness_test
python -m unittest discover -s scripts -p "test_*.py"
python scripts/validate_workspace.py

Verification Notes

Confirm RED failed before production code was added.
Confirm no parser, assembly, or results writer code was changed.
Update phases/euler-beam-3d/index.json step 7:
- success: "status": "completed", "summary": "local 3D Euler beam stiffness and local end-force kernel added"
- failure after retries: "status": "error", "error_message": "<specific error>"
- blocked: "status": "blocked", "blocked_reason": "<specific reason>"

Forbidden

Do not add shear deformation or Timoshenko terms.
Do not modify reference artifacts.

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