Step 8: global-transform-recovery

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

Task

Use TDD to add local/global transformation and global end-force recovery to the 3D Euler-Bernoulli beam kernel.

Extend the API in /src/fesa/elements/euler_beam_3d.hpp:

using Vector3 = std::array<double, 3>;

struct EulerBeam3DGeometry {
    Vector3 node1;
    Vector3 node2;
    Vector3 orientation;
};

Matrix12 euler_beam_3d_global_stiffness(const EulerBeam3DGeometry& geometry,
                                        const EulerBeam3DSection& section);
Vector12 euler_beam_3d_global_end_forces(const EulerBeam3DGeometry& geometry,
                                         const EulerBeam3DSection& section,
                                         const Vector12& global_displacements);

Implementation rules:

local x is normalized node2 - node1
local y is the normalized projection of orientation onto the plane normal to local x
local z = x cross y
use the same 3x3 rotation block for translational and rotational DOFs
compute K_global = T^T K_local T
compute global end forces by transforming global displacements to local, recovering local forces, then transforming forces back to global
throw std::invalid_argument for zero-length element, zero orientation vector, or orientation parallel to the beam axis

Tests To Write First

Create /tests/unit/euler_beam_3d_transform_recovery_test.cpp before production code.

Test behavior:

Axis-aligned beam from (0,0,0) to (2,0,0) with orientation (0,1,0) gives global stiffness equal to local stiffness.
A rotated beam preserves stiffness symmetry.
A rigid global translation vector produces near-zero global end forces.
A simple global axial extension on the axis-aligned beam produces equal and opposite axial end forces.
Parallel orientation vector throws std::invalid_argument.

RED command:

ctest --test-dir build/msvc-debug --output-on-failure -C Debug -R euler_beam_3d_transform_recovery_test

Expected RED: test executable missing or compile failure because the transform 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_transform_recovery_test

Acceptance Criteria

ctest --test-dir build/msvc-debug --output-on-failure -C Debug -R "euler_beam_3d_(local_stiffness|transform_recovery)_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 local stiffness tests still pass.
Update phases/euler-beam-3d/index.json step 8:
- success: "status": "completed", "summary": "global transform and global end-force recovery added for 3D Euler beam"
- failure after retries: "status": "error", "error_message": "<specific error>"
- blocked: "status": "blocked", "blocked_reason": "<specific reason>"

Forbidden

Do not add assembly or solver integration in this step.
Do not modify reference artifacts.

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