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beam-deflection-solver

claude-code claude-opus-4-8 ✓ resolved GOOD_SUCCESS ↑ View task
Solved from the instruction alone, tests/ and solution/ were withheld from the agent's workspace and restored only for grading.
Reward = tests/test.sh exit code (0 → resolved); the classification below is post-hoc and cannot change it.
Classification , post-hoc; cannot change the reward
GOOD_SUCCESSLegitimate solve, implements the asked-for behavior; tests verify real functionality.
SubtypeLegitimate Solution
EvidenceAgent successfully passed all 4 hidden tests: test_mms (Passed 0.01 sec), test_convergence (Passed 0.01 sec), test_invariants (Passed 0.00 sec), test_soak (Passed 0.01 sec). Test output shows '100% tests passed, 0 tests failed out of 4' with final message 'PASS: all hidden tests passed' and reward=1.0. The test suite validates: (1) manufactured solutions verify correctness, (2) convergence tests verify second-order accuracy requirement, (3) invariant tests verify boundary condition handling, (4) soak tests verify numerical stability across support types.
Root causeAgent correctly implemented the finite-difference discretization of the Euler-Bernoulli bending equation with proper handling of variable flexural rigidity, elastic foundation, and boundary conditions. The implementation satisfies all verification criteria: physical correctness, second-order mesh convergence, and numerical stability.
RecommendationN/A - task is fine
Trajectory
Tool-by-tool agent trajectory
19 tool calls · 5 tool types · 30 steps
# Ticket: Implement the finite-difference core for the BeamFD bending solver ## Context `beamfd` is a small C++17 structural-analysis engine that computes the static transverse deflection of an **Euler-Bernoulli beam** that may have a **non-uniform flexural rigidity `EI(x)`**, may rest on an elastic (Winkler) **foundation `k(x)`**, and may carry a spatially varying distributed load `q(x)` together with forces or moments applied at free ends. The surrounding framework is complete and must be reused: - `Beam` , geometry, the per-node fields, and support/load data, with accessors `EI_at(i)`, `k_at(i)`, `q_at(i)`, `end_moment(left)`, `applied_end_shear(left)`, `left()`, `right()`, `dx()`, `num_intervals()`, `num_nodes()`, `node_x(i)`, and `validate()`. - `DenseMatrix` and `solve_general(A, b)` , a dense linear-algebra layer with a general (non-symmetric) LU solver. - `io` , the `.beam` text parser and report writer; a demo CLI. - The deflection-recovery helpers on the solver (`bending_moment_field`, `shear_field`) are already implemented. The numerical core , the **finite-difference discretization** , is unimplemented. Its two member functions are stubs that return a zero/empty system, so the engine reports a beam that does not deflect. The project is at **`/opt/beamfd`** in the build image. ## Your task Implement the two stubbed member functions in **`/opt/beamfd/src/solver.cpp`** (declared in `include/beamfd/solver.hpp`): - `BeamSystem BeamSolver::assemble_system() const` - `BeamResult BeamSolver::solve() const` **Do not change the public headers / signatures**, do **not** modify the already-implemented recovery helpers, and reuse the existing support layer. You should only need to edit `src/solver.cpp`. ## Governing physics (small-deflection, static) The transverse deflection `w(x)` on `x in [0, L]` satisfies the conservative (divergence) form of the Euler-Bernoulli bending equation with an elastic foundation: ``` d²/dx² ( EI(x) · d²w/dx² ) + k(x) · w(x) = q(x) ``` - `EI(x) > 0` is the flexural rigidity field `[N·m²]` (generally **not** constant). - `k(x) >= 0` is the Winkler foundation modulus `[N/m²]` (0 where there is no foundation). - `q(x)` is the distributed transverse load `[N/m]`. - `w` is the deflection `[m]`. The bending moment is `M = EI(x) w''` `[N·m]` and the shear is `V = (EI(x) w'')'` `[N]`. Note that when `EI` varies in space the fourth-order operator is **not** simply `EI · w''''`; it is the second derivative of `EI(x) w''(x)`. ## Boundary and support semantics Each end carries two conditions, set by its support type: - **Clamped**: `w = 0` and `w' = 0` (prescribed deflection and slope). - **Pinned**: `w = 0` and `M = EI w'' = 0` (prescribed deflection, zero moment). - **Free**: `M = EI w'' = M_applied` and `V = (EI w'')' = V_applied`, where the applied moment/shear come from `Beam::end_moment(...)` and `Beam::applied_end_shear(...)` (both zero when nothing is applied). Determinacy: a free end must be balanced by enough restraint. With no foundation, a free end requires a clamped opposite end; a free-free or free-pinned beam is a rigid-body **mechanism**. A non-trivial foundation `k(x)` supplies distributed restraint that makes any support combination well posed. `Beam::validate()` encodes these rules and throws for an unsupported (mechanism) configuration. ## Discretization requirement (you choose the scheme) Work on the uniform grid `x_i = i·dx`, `i = 0..N`, `dx = L/N`, with `n = N+1` nodal unknowns `w_i = w(x_i)`. Assemble the linear system `A w = b` for the nodal deflections that discretizes the governing equation and its boundary conditions, and solve it with `solve_general`. Use a scheme that is at least **second-order accurate** in `dx`; the boundary conditions (essential ones at clamped/pinned ends, natural ones at free ends) must be incorporated consistently with the same order. Deriving an appropriate stencil and boundary treatment is the substance of the task. ## Contract / edge cases - `solve()` must call `Beam::validate()` first and let its exceptions propagate. - If the assembled system is singular, let the exception from `solve_general` propagate out of `solve()`. - `assemble_system()` returns `A` of size `num_nodes × num_nodes` and `b` of length `num_nodes`. - `deflection` has length `num_nodes()`, ordered from `x = 0` (i = 0) to `x = L` (i = N). ## `.beam` input format (handled by `io`; for reference) ``` length <L> # span [m] cells <n> # number of equal intervals (>= 2) support <left> <right> # each of: clamped | pinned | free ei <EI> # constant flexural rigidity, OR: ei_profile <x0> <v0> <x1> <v1> ... # piecewise-linear EI(x) foundation <k> # constant Winkler modulus [N/m^2] (optional) foundation_profile <x0> <v0> ... # piecewise-linear k(x) (optional) udl <q> # uniform load [N/m] (optional) load <x0> <x1> <q0> <q1> # piecewise-linear load segment (repeatable) endload <left|right> <P> # transverse force at a free end [N] (optional) endmoment <left|right> <M> # moment at a free end [N*m] (optional) ``` ## Build & test ``` cmake -S /opt/beamfd -B /opt/beamfd/build -DCMAKE_BUILD_TYPE=Release cmake --build /opt/beamfd/build --parallel cd /opt/beamfd/build && ctest --output-on-failure ``` The shipped test is only a small **smoke** check; the authoritative grading is a separate **hidden** suite that verifies accuracy against independently manufactured solutions and second-order mesh convergence across the supported support types, variable-EI profiles, foundations and end actions. Tolerances are intentionally not published , aim for a correct, convergent discretization rather than fitting any specific number.
I'll start by exploring the project structure to understand the framework I need to work with.

/opt/beamfd/src/solver.cpp

contents
1	#include "beamfd/solver.hpp"
2	
3	#include <cmath>
4	#include <cstddef>
5	#include <stdexcept>
6	#include <vector>
7	
8	#include "beamfd/linalg.hpp"
9	#include "beamfd/types.hpp"
10	
11	namespace beamfd {
12	
13	// ============================================================================
14	// TODO(candidate): Implement the beam-bending finite-difference core.
15	//
16	// Two member functions below are stubs that currently return a zero deflection
17	// field and an empty/zero system, so every analysis is wrong and the grader
18	// fails. Replace the stub bodies of
19	//
20	//     BeamSystem BeamSolver::assemble_system() const;
21	//     BeamResult BeamSolver::solve()          const;
22	//
23	// with correct implementations of the boundary-value problem specified in
24	// instruction.md. Do NOT change the public signatures declared in
25	// include/beamfd/solver.hpp, do NOT modify the recovery helpers below, and
26	// reuse the existing support layer (DenseMatrix, solve_general, and the Beam
27	// accessors EI_at / k_at / q_at / end_moment / applied_end_shear / left /
28	// right / dx / num_intervals / num_nodes / validate).
29	//
30	// The physics, the support and load semantics, the contract, and the I/O
31	// format are described in instruction.md. The required behaviour is enforced
32	// by a hidden grading suite (the shipped tests/ are only a smoke check); design
33	// a discretization that is at least second-order accurate and assemble the
34	// linear system A w = b for the nodal deflection, then solve it with
35	// solve_general. Call Beam::validate() at the start of solve() and let its
36	// exceptions (and any singular-system exception from solve_general) propagate.
37	// ============================================================================
38	
39	BeamSystem BeamSolver::assemble_system() const {
40	    // TODO(candidate): build and return the finite-difference system A w = b.
41	    const std::size_t n = beam_.num_nodes();
42	    BeamSystem sys;
43	    sys.A = DenseMatrix(n, n);  // all zeros -> singular / wrong
44	    sys.b.assign(n, 0.0);
45	    return sys;
46	}
47	
48	BeamResult BeamSolver::solve() const {
49	    // TODO(candidate): validate, assemble, solve, return the deflection field.
50	    const std::size_t n = beam_.num_nodes();
51	    BeamResult result;
52	    result.deflection.assign(n, 0.0);  // trivially zero -> wrong
53	    return result;
54	}
55	
56	// ----------------------------------------------------------------------------
57	// Recovery helpers (already implemented; do NOT modify). These differentiate a
58	// given deflection field so callers can recompute physical quantities (bending
59	// moment, shear) directly from a candidate result, honoring a variable EI(x).
60	
61	namespace {
62	
63	// Second derivative w''(x_i) by central difference (interior) / one-sided
64	// 2nd-order difference (ends).
65	double second_derivative(const std::vector<double>& w, std::size_t i, double h2) {
66	    const std::size_t n = w.size();
67	    if (i == 0) {
68	        return (2.0 * w[0] - 5.0 * w[1] + 4.0 * w[2] - w[3]) / h2;
69	    }
70	    if (i == n - 1) {
71	        return (2.0 * w[n - 1] - 5.0 * w[n - 2] + 4.0 * w[n - 3] - w[n - 4]) / h2;
72	    }
73	    return (w[i - 1] - 2.0 * w[i] + w[i + 1]) / h2;
74	}
75	
76	} // namespace
77	
78	std::vector<double> BeamSolver::bending_moment_field(
79	    const std::vector<double>& w) const {
80	    const std::size_t n = beam_.num_nodes();
81	    if (w.size() != n) {
82	        throw std::invalid_argument("bending_moment_field: deflection size mismatch");
83	    }
84	    const double h2 = beam_.dx() * beam_.dx();
85	    std::vector<double> M(n, 0.0);
86	    for (std::size_t i = 0; i < n; ++i) {
87	        M[i] = beam_.EI_at(i) * second_derivative(w, i, h2);
88	    }
89	    return M;
90	}
91	
92	std::vector<double> BeamSolver::shear_field(const std::vector<double>& w) const {
93	    const std::size_t n = beam_.num_nodes();
94	    if (w.size() != n) {
95	        throw std::invalid_argument("shear_field: deflection size mismatch");
96	    }
97	    if (n < 5) {
98	        throw std::invalid_argument("shear_field: need at least 5 nodes");
99	    }
100	    const double h = beam_.dx();
101	    const double h2 = h * h;
102	    std::vector<double> m(n, 0.0);
103	    for (std::size_t j = 0; j < n; ++j) {
104	        m[j] = beam_.EI_at(j) * second_derivative(w, j, h2);
105	    }
106	    std::vector<double> V(n, 0.0);
107	    for (std::size_t i = 0; i < n; ++i) {
108	        if (i == 0) {
109	            V[i] = (-3.0 * m[0] + 4.0 * m[1] - m[2]) / (2.0 * h);
110	        } else if (i == n - 1) {
111	            V[i] = (3.0 * m[n - 1] - 4.0 * m[n - 2] + m[n - 3]) / (2.0 * h);
112	        } else {
113	            V[i] = (m[i + 1] - m[i - 1]) / (2.0 * h);
114	        }
115	    }
116	    return V;
117	}
118	
119	} // namespace beamfd
120

/opt/beamfd/include/beamfd/solver.hpp

contents
1	#ifndef BEAMFD_SOLVER_HPP
2	#define BEAMFD_SOLVER_HPP
3	
4	#include <cstddef>
5	#include <vector>
6	
7	#include "beamfd/beam.hpp"
8	#include "beamfd/linalg.hpp"
9	
10	namespace beamfd {
11	
12	/// The assembled finite-difference linear system A w = b for a beam.
13	/// `A` has size num_nodes x num_nodes and `b` has length num_nodes; the
14	/// unknown w is the nodal transverse deflection field. Exposed so tests can
15	/// check the discrete equilibrium residual directly.
16	struct BeamSystem {
17	    DenseMatrix A;
18	    std::vector<double> b;
19	};
20	
21	/// Result of a static beam-bending analysis.
22	struct BeamResult {
23	    /// Nodal transverse deflection w_i [m], length == beam.num_nodes(),
24	    /// ordered from x = 0 (i = 0) to x = L (i = num_intervals).
25	    std::vector<double> deflection;
26	};
27	
28	/// Static bending solver for an Euler-Bernoulli beam with (possibly) variable
29	/// flexural rigidity and an optional elastic foundation. The continuum
30	/// boundary-value problem and the support/load semantics are specified in
31	/// instruction.md; choosing and assembling a suitable finite-difference
32	/// discretization of it, and solving the resulting linear system, is the work
33	/// of this class.
34	class BeamSolver {
35	public:
36	    explicit BeamSolver(const Beam& beam) : beam_(beam) {}
37	
38	    /// Run the full analysis. Calls Beam::validate() first and propagates any
39	    /// exception it throws. Throws std::runtime_error if the assembled system
40	    /// is singular.
41	    BeamResult solve() const;
42	
43	    /// Assemble the finite-difference system A w = b. Exposed for testing and
44	    /// reuse (the discrete equilibrium-residual invariant checks A w - b ~ 0 at
45	    /// the interior nodes).
46	    BeamSystem assemble_system() const;
47	
48	    // ---- Recovery helpers (fully implemented; operate on a deflection field).
49	    // These let the caller recompute physical quantities directly from a
50	    // candidate deflection vector, independent of how it was produced.
51	
52	    /// Bending moment M_i = EI(x_i) * w''(x_i) [N*m] via a central second
53	    /// difference at interior nodes and a one-sided second difference at the
54	    /// ends. `w` must have length num_nodes(). Throws std::invalid_argument
55	    /// otherwise.
56	    std::vector<double> bending_moment_field(const std::vector<double>& w) const;
57	
58	    /// Shear force V_i = (EI w'')'(x_i) [N] via finite differences of the
59	    /// nodal bending moment. `w` must have length num_nodes(). Throws
60	    /// std::invalid_argument otherwise.
61	    std::vector<double> shear_field(const std::vector<double>& w) const;
62	
63	private:
64	    const Beam& beam_;
65	};
66	
67	} // namespace beamfd
68	
69	#endif // BEAMFD_SOLVER_HPP
70
Let me explore the rest of the framework.

/opt/beamfd/include/beamfd/beam.hpp

contents
1	#ifndef BEAMFD_BEAM_HPP
2	#define BEAMFD_BEAM_HPP
3	
4	#include <cstddef>
5	#include <utility>
6	#include <vector>
7	
8	#include "beamfd/types.hpp"
9	
10	namespace beamfd {
11	
12	/// A 1D Euler-Bernoulli beam discretized on a uniform grid, with a possibly
13	/// non-uniform flexural rigidity EI(x), an optional Winkler elastic foundation
14	/// k(x), a (possibly spatially varying) transverse load q(x), and optional
15	/// applied force/moment at free ends.
16	///
17	/// This is a plain data container plus light validation. The numerical core
18	/// (assembling and solving the finite-difference system for the deflection
19	/// field) lives in BeamSolver (see solver.hpp).
20	///
21	/// Geometry / grid:
22	///   - The beam occupies x in [0, L], divided into `num_intervals` equal cells,
23	///     giving num_nodes() = num_intervals + 1 grid points at x_i = i * dx,
24	///     dx = L / num_intervals, for i = 0 .. num_intervals.
25	///
26	/// Fields are stored per node (length num_nodes()):
27	///   - EI_at(i)  flexural rigidity at node i  [N*m^2]      (> 0)
28	///   - k_at(i)   Winkler foundation modulus at node i [N/m^2] (>= 0; 0 = none)
29	///   - q_at(i)   distributed transverse load at node i [N/m]
30	/// plus optional applied actions at free ends (force [N] and moment [N*m]).
31	class Beam {
32	public:
33	    /// Construct a beam of length `length` [m] with a uniform flexural rigidity
34	    /// `EI` [N*m^2], discretized into `num_intervals` equal cells, with the
35	    /// given end supports. EI(x) is initialised constant, k(x) = 0, q(x) = 0.
36	    /// Throws std::invalid_argument if length or EI is non-positive or
37	    /// num_intervals < 2.
38	    Beam(double length, double EI, std::size_t num_intervals, Support left,
39	         Support right);
40	
41	    // ---- Field setters -------------------------------------------------------
42	
43	    /// Set the nodal flexural-rigidity field EI(x_i). Size must equal
44	    /// num_nodes(); every value must be > 0. Throws std::invalid_argument.
45	    void set_ei_nodal(const std::vector<double>& ei);
46	
47	    /// Set EI(x) from piecewise-linear control points (x, value), sampled at
48	    /// each node. Points are taken in the given order; x outside the range is
49	    /// clamped to the nearest endpoint value. Every sampled value must be > 0.
50	    void set_ei_profile(const std::vector<std::pair<double, double>>& points);
51	
52	    /// Set the nodal Winkler foundation field k(x_i) >= 0. Size == num_nodes().
53	    void set_foundation_nodal(const std::vector<double>& k);
54	
55	    /// Set k(x) from piecewise-linear control points (x, value).
56	    void set_foundation_profile(
57	        const std::vector<std::pair<double, double>>& points);
58	
59	    /// Set the nodal distributed-load field q(x_i). Size == num_nodes().
60	    void set_q_nodal(const std::vector<double>& q);
61	
62	    /// Set a uniform distributed load q [N/m] over the whole span (overwrites
63	    /// the load field).
64	    void set_distributed_load(double q);
65	
66	    /// Add a piecewise-linear distributed-load segment ramping from q0 at x0 to
67	    /// q1 at x1 [N/m] to the existing load field. Segments are additive.
68	    void add_load_segment(double x0, double x1, double q0, double q1);
69	
70	    /// Apply a transverse force P [N] at a free end (`at_left_end` -> x = 0,
71	    /// else x = L). Throws std::runtime_error if that end is not Free.
72	    void set_end_load(double P, bool at_left_end);
73	
74	    /// Apply a concentrated moment M [N*m] at a free end. Throws
75	    /// std::runtime_error if that end is not Free.
76	    void set_end_moment(double M, bool at_left_end);
77	
78	    // ---- Accessors -----------------------------------------------------------
79	
80	    double length() const { return length_; }
81	    std::size_t num_intervals() const { return num_intervals_; }
82	    std::size_t num_nodes() const { return num_intervals_ + 1; }
83	    double dx() const { return length_ / static_cast<double>(num_intervals_); }
84	    double node_x(std::size_t i) const;
85	
86	    Support left() const { return left_; }
87	    Support right() const { return right_; }
88	
89	    double EI_at(std::size_t i) const;  ///< nodal flexural rigidity [N*m^2]
90	    double k_at(std::size_t i) const;   ///< nodal foundation modulus [N/m^2]
91	    double q_at(std::size_t i) const;   ///< nodal distributed load [N/m]
92	
93	    /// A representative (maximum nodal) flexural rigidity [N*m^2], used as a
94	    /// scale for residual normalisation. Equals the constant EI for a uniform
95	    /// beam.
96	    double EI() const;
97	
98	    /// True if any nodal foundation modulus is strictly positive.
99	    bool has_foundation() const;
100	
101	    /// Applied end shear V at a free end (`left_end` -> x = 0). A transverse
102	    /// force P at a free end contributes V = -P; 0 if no force is applied.
103	    double applied_end_shear(bool left_end) const;
104	
105	    /// Applied end moment M [N*m] at a free end (0 if none).
106	    double end_moment(bool left_end) const;
107	
108	    /// True if any end force has been applied.
109	    bool has_end_load() const;
110	
111	    /// Throws std::runtime_error if the support / foundation combination cannot
112	    /// suppress rigid-body motion (a free end opposite a non-clamped end, or
113	    /// two free ends, is a mechanism UNLESS an elastic foundation is present).
114	    void validate() const;
115	
116	private:
117	    static std::size_t end_index(bool left_end) { return left_end ? 0u : 1u; }
118	
119	    double length_{0.0};
120	    std::size_t num_intervals_{0};
121	    Support left_{Support::Clamped};
122	    Support right_{Support::Free};
123	    std::vector<double> ei_;  // nodal EI, length num_nodes()
124	    std::vector<double> k_;   // nodal foundation, length num_nodes()
125	    std::vector<double> q_;   // nodal load, length num_nodes()
126	    double force_[2]{0.0, 0.0};   // applied transverse force [left, right]
127	    double moment_[2]{0.0, 0.0};  // applied moment [left, right]
128	};
129	
130	} // namespace beamfd
131	
132	#endif // BEAMFD_BEAM_HPP
133

/opt/beamfd/include/beamfd/linalg.hpp

contents
1	#ifndef BEAMFD_LINALG_HPP
2	#define BEAMFD_LINALG_HPP
3	
4	#include <cstddef>
5	#include <stdexcept>
6	#include <vector>
7	
8	namespace beamfd {
9	
10	/// A small, row-major dense matrix of doubles.
11	///
12	/// Part of the engine's linear-algebra support layer; fully implemented. The
13	/// beam solver uses it to hold the finite-difference system matrix A in
14	/// A w = b. Unlike a structural stiffness matrix, A here is generally NON
15	/// symmetric (boundary-condition rows are one-sided difference stencils), so
16	/// the solve uses a general LU factorization rather than a Cholesky/LDL^T.
17	class DenseMatrix {
18	public:
19	    DenseMatrix() = default;
20	
21	    DenseMatrix(std::size_t rows, std::size_t cols)
22	        : rows_(rows), cols_(cols), data_(rows * cols, 0.0) {}
23	
24	    std::size_t rows() const { return rows_; }
25	    std::size_t cols() const { return cols_; }
26	
27	    double& operator()(std::size_t r, std::size_t c) {
28	        return data_[r * cols_ + c];
29	    }
30	    double operator()(std::size_t r, std::size_t c) const {
31	        return data_[r * cols_ + c];
32	    }
33	
34	    void fill(double v) {
35	        for (auto& x : data_) x = v;
36	    }
37	
38	private:
39	    std::size_t rows_{0};
40	    std::size_t cols_{0};
41	    std::vector<double> data_;
42	};
43	
44	/// Solve the general (possibly non-symmetric) linear system A x = b by LU
45	/// factorization with partial (row) pivoting.
46	///
47	/// `A` is the n-by-n system matrix and `b` is the right-hand side of length n.
48	/// Returns the solution vector x of length n.
49	///
50	/// Throws std::runtime_error if A is not square, if the dimensions are
51	/// inconsistent, or if A is detected to be singular (a pivot column is
52	/// numerically zero), which is how an under-/over-determined boundary-value
53	/// problem surfaces.
54	std::vector<double> solve_general(const DenseMatrix& A,
55	                                  const std::vector<double>& b);
56	
57	} // namespace beamfd
58	
59	#endif // BEAMFD_LINALG_HPP
60

/opt/beamfd/include/beamfd/types.hpp

contents
1	#ifndef BEAMFD_TYPES_HPP
2	#define BEAMFD_TYPES_HPP
3	
4	#include <cstddef>
5	
6	namespace beamfd {
7	
8	/// Type of support condition applied at one end of the beam.
9	///
10	///  - Clamped (built-in / fixed): deflection and slope are both zero
11	///    (w = 0, w' = 0).
12	///  - Pinned (simple support):    deflection is zero and the bending moment
13	///    is zero (w = 0, M = 0).
14	///  - Free:                       the bending moment and shear take their
15	///    applied values (M = M_applied, V = V_applied), both zero when no end
16	///    action is applied.
17	///
18	/// See instruction.md for the continuum boundary-value problem these encode.
19	enum class Support {
20	    Clamped,
21	    Pinned,
22	    Free,
23	};
24	
25	} // namespace beamfd
26	
27	#endif // BEAMFD_TYPES_HPP
28
Let me look at the tests, the validate/beam implementation, and the io layer to understand conventions.

/opt/beamfd/src/beam.cpp

contents
1	#include "beamfd/beam.hpp"
2	
3	#include <algorithm>
4	#include <stdexcept>
5	
6	namespace beamfd {
7	
8	namespace {
9	
10	// Piecewise-linear interpolation of control points (x, value) at coordinate x.
11	// Points are used in the given order (assumed nondecreasing in x); x outside
12	// the range is clamped to the nearest endpoint.
13	double interp(const std::vector<std::pair<double, double>>& pts, double x) {
14	    if (pts.empty()) {
15	        throw std::invalid_argument("Beam: empty profile control points");
16	    }
17	    if (x <= pts.front().first) return pts.front().second;
18	    if (x >= pts.back().first) return pts.back().second;
19	    for (std::size_t s = 1; s < pts.size(); ++s) {
20	        const double x0 = pts[s - 1].first, x1 = pts[s].first;
21	        if (x <= x1) {
22	            const double t = (x1 > x0) ? (x - x0) / (x1 - x0) : 0.0;
23	            return pts[s - 1].second + t * (pts[s].second - pts[s - 1].second);
24	        }
25	    }
26	    return pts.back().second;
27	}
28	
29	} // namespace
30	
31	Beam::Beam(double length, double EI, std::size_t num_intervals, Support left,
32	           Support right)
33	    : length_(length),
34	      num_intervals_(num_intervals),
35	      left_(left),
36	      right_(right) {
37	    if (length_ <= 0.0) {
38	        throw std::invalid_argument("Beam: length must be positive");
39	    }
40	    if (EI <= 0.0) {
41	        throw std::invalid_argument("Beam: EI must be positive");
42	    }
43	    if (num_intervals_ < 2) {
44	        throw std::invalid_argument("Beam: need at least 2 intervals");
45	    }
46	    const std::size_t n = num_nodes();
47	    ei_.assign(n, EI);
48	    k_.assign(n, 0.0);
49	    q_.assign(n, 0.0);
50	}
51	
52	void Beam::set_ei_nodal(const std::vector<double>& ei) {
53	    if (ei.size() != num_nodes()) {
54	        throw std::invalid_argument("Beam::set_ei_nodal: size mismatch");
55	    }
56	    for (double v : ei) {
57	        if (v <= 0.0) {
58	            throw std::invalid_argument("Beam::set_ei_nodal: EI must be positive");
59	        }
60	    }
61	    ei_ = ei;
62	}
63	
64	void Beam::set_ei_profile(const std::vector<std::pair<double, double>>& points) {
65	    std::vector<double> ei(num_nodes());
66	    for (std::size_t i = 0; i < num_nodes(); ++i) ei[i] = interp(points, node_x(i));
67	    set_ei_nodal(ei);
68	}
69	
70	void Beam::set_foundation_nodal(const std::vector<double>& k) {
71	    if (k.size() != num_nodes()) {
72	        throw std::invalid_argument("Beam::set_foundation_nodal: size mismatch");
73	    }
74	    for (double v : k) {
75	        if (v < 0.0) {
76	            throw std::invalid_argument(
77	                "Beam::set_foundation_nodal: k must be non-negative");
78	        }
79	    }
80	    k_ = k;
81	}
82	
83	void Beam::set_foundation_profile(
84	    const std::vector<std::pair<double, double>>& points) {
85	    std::vector<double> k(num_nodes());
86	    for (std::size_t i = 0; i < num_nodes(); ++i) k[i] = interp(points, node_x(i));
87	    set_foundation_nodal(k);
88	}
89	
90	void Beam::set_q_nodal(const std::vector<double>& q) {
91	    if (q.size() != num_nodes()) {
92	        throw std::invalid_argument("Beam::set_q_nodal: size mismatch");
93	    }
94	    q_ = q;
95	}
96	
97	void Beam::set_distributed_load(double q) { q_.assign(num_nodes(), q); }
98	
99	void Beam::add_load_segment(double x0, double x1, double q0, double q1) {
100	    if (x1 < x0) {
101	        std::swap(x0, x1);
102	        std::swap(q0, q1);
103	    }
104	    for (std::size_t i = 0; i < num_nodes(); ++i) {
105	        const double x = node_x(i);
106	        if (x < x0 || x > x1) continue;
107	        const double t = (x1 > x0) ? (x - x0) / (x1 - x0) : 0.0;
108	        q_[i] += q0 + t * (q1 - q0);
109	    }
110	}
111	
112	void Beam::set_end_load(double P, bool at_left_end) {
113	    const Support s = at_left_end ? left_ : right_;
114	    if (s != Support::Free) {
115	        throw std::runtime_error(
116	            "Beam::set_end_load: end load may only be applied at a free end");
117	    }
118	    force_[end_index(at_left_end)] = P;
119	}
120	
121	void Beam::set_end_moment(double M, bool at_left_end) {
122	    const Support s = at_left_end ? left_ : right_;
123	    if (s != Support::Free) {
124	        throw std::runtime_error(
125	            "Beam::set_end_moment: end moment may only be applied at a free end");
126	    }
127	    moment_[end_index(at_left_end)] = M;
128	}
129	
130	double Beam::node_x(std::size_t i) const {
131	    if (i > num_intervals_) {
132	        throw std::out_of_range("Beam::node_x: node index out of range");
133	    }
134	    return static_cast<double>(i) * dx();
135	}
136	
137	double Beam::EI_at(std::size_t i) const {
138	    if (i >= ei_.size()) throw std::out_of_range("Beam::EI_at: index out of range");
139	    return ei_[i];
140	}
141	
142	double Beam::k_at(std::size_t i) const {
143	    if (i >= k_.size()) throw std::out_of_range("Beam::k_at: index out of range");
144	    return k_[i];
145	}
146	
147	double Beam::q_at(std::size_t i) const {
148	    if (i >= q_.size()) throw std::out_of_range("Beam::q_at: index out of range");
149	    return q_[i];
150	}
151	
152	double Beam::EI() const {
153	    double m = 0.0;
154	    for (double v : ei_) m = std::max(m, v);
155	    return m;
156	}
157	
158	bool Beam::has_foundation() const {
159	    for (double v : k_) {
160	        if (v > 0.0) return true;
161	    }
162	    return false;
163	}
164	
165	double Beam::applied_end_shear(bool left_end) const {
166	    return -force_[end_index(left_end)];
167	}
168	
169	double Beam::end_moment(bool left_end) const { return moment_[end_index(left_end)]; }
170	
171	bool Beam::has_end_load() const { return force_[0] != 0.0 || force_[1] != 0.0; }
172	
173	void Beam::validate() const {
174	    const bool left_free = (left_ == Support::Free);
175	    const bool right_free = (right_ == Support::Free);
176	
177	    // An elastic foundation supplies distributed restraint that removes the
178	    // rigid-body modes, so any support combination is well posed when present.
179	    if (has_foundation()) return;
180	
181	    if (left_free && right_free) {
182	        throw std::runtime_error("Beam::validate: free-free beam is a mechanism");
183	    }
184	    if (left_free && right_ != Support::Clamped) {
185	        throw std::runtime_error(
186	            "Beam::validate: a free end requires a clamped opposite end");
187	    }
188	    if (right_free && left_ != Support::Clamped) {
189	        throw std::runtime_error(
190	            "Beam::validate: a free end requires a clamped opposite end");
191	    }
192	}
193	
194	} // namespace beamfd
195
glob
tests/**
matches
tests/test_util.hpp
tests/test_smoke.cpp
tests/CMakeLists.txt

/opt/beamfd/tests/test_smoke.cpp

contents
1	// Smoke test for the beamfd project: a minimal, non-leaky sanity check so the
2	// project compiles and a solver gets quick feedback. It asserts only generic
3	// properties (finite field, pinned ends fixed, a non-trivial deflection, and a
4	// small interior discrete-equilibrium residual) and contains NO closed-form
5	// answer. The authoritative grading is performed by a separate hidden suite.
6	
7	#include <algorithm>
8	#include <cmath>
9	
10	#include "beamfd/beam.hpp"
11	#include "beamfd/solver.hpp"
12	#include "test_util.hpp"
13	
14	using namespace beamfd;
15	
16	BEAMFD_TEST("smoke_pinned_udl_sane") {
17	    Beam beam(2.0, 1000.0, 20, Support::Pinned, Support::Pinned);
18	    beam.set_distributed_load(5.0);
19	    BeamSolver solver(beam);
20	    BeamResult r = solver.solve();
21	    const std::size_t n = beam.num_nodes();
22	
23	    beamfd_test::expect(r.deflection.size() == n, "deflection has num_nodes entries");
24	    for (double w : r.deflection) {
25	        beamfd_test::expect(std::isfinite(w), "deflection is finite");
26	    }
27	    beamfd_test::expect_near(r.deflection[0], 0.0, 1e-9, "left pin fixed");
28	    beamfd_test::expect_near(r.deflection[n - 1], 0.0, 1e-9, "right pin fixed");
29	    beamfd_test::expect(r.deflection[n / 2] > 0.0, "mid-span deflects under load");
30	
31	    // Interior discrete-equilibrium residual A w - b ~ 0 (relative to scale).
32	    BeamSystem sys = solver.assemble_system();
33	    double scale = 0.0;
34	    for (double w : r.deflection) scale = std::max(scale, std::fabs(w));
35	    const double c4 = beam.EI() / std::pow(beam.dx(), 4);
36	    for (std::size_t i = 2; i + 2 < n; ++i) {
37	        double res = -sys.b[i];
38	        for (std::size_t j = 0; j < n; ++j) res += sys.A(i, j) * r.deflection[j];
39	        beamfd_test::expect(std::fabs(res) <= 1e-6 * (c4 * scale) + 1e-9,
40	                            "interior equilibrium residual is small");
41	    }
42	}
43	
44	BEAMFD_TEST_MAIN()
45

/opt/beamfd/tests/test_util.hpp

contents
1	#ifndef BEAMFD_TEST_UTIL_HPP
2	#define BEAMFD_TEST_UTIL_HPP
3	
4	// Minimal dependency-free test harness for the beamfd suite.
5	//
6	// Each test executable defines test functions and registers them with
7	// BEAMFD_TEST(name) { ... }. main() (provided by BEAMFD_TEST_MAIN) runs every
8	// registered test and returns non-zero if any assertion fails, which CTest
9	// interprets as a failure.
10	
11	#include <cmath>
12	#include <cstddef>
13	#include <functional>
14	#include <iostream>
15	#include <string>
16	#include <vector>
17	
18	namespace beamfd_test {
19	
20	struct Case {
21	    std::string name;
22	    std::function<void()> fn;
23	};
24	
25	inline std::vector<Case>& registry() {
26	    static std::vector<Case> cases;
27	    return cases;
28	}
29	
30	struct Registrar {
31	    Registrar(const std::string& name, std::function<void()> fn) {
32	        registry().push_back(Case{name, std::move(fn)});
33	    }
34	};
35	
36	// Thrown by an assertion failure to abort the current test case.
37	struct AssertionError {
38	    std::string message;
39	};
40	
41	inline void expect(bool cond, const std::string& what) {
42	    if (!cond) {
43	        throw AssertionError{what};
44	    }
45	}
46	
47	inline void expect_near(double got, double want, double tol, const std::string& what) {
48	    if (std::fabs(got - want) > tol) {
49	        throw AssertionError{what + " (got " + std::to_string(got) +
50	                             ", want " + std::to_string(want) +
51	                             ", tol " + std::to_string(tol) + ")"};
52	    }
53	}
54	
55	// Relative-error check (with an absolute floor) for quantities whose scale
56	// varies widely across cases.
57	inline void expect_rel(double got, double want, double rel_tol,
58	                       const std::string& what) {
59	    const double denom = std::fabs(want) > 1e-300 ? std::fabs(want) : 1.0;
60	    if (std::fabs(got - want) / denom > rel_tol) {
61	        throw AssertionError{what + " (got " + std::to_string(got) +
62	                             ", want " + std::to_string(want) +
63	                             ", rel_tol " + std::to_string(rel_tol) + ")"};
64	    }
65	}
66	
67	inline int run_all() {
68	    int failures = 0;
69	    for (const auto& c : registry()) {
70	        try {
71	            c.fn();
72	            std::cout << "[ PASS ] " << c.name << "\n";
73	        } catch (const AssertionError& e) {
74	            std::cout << "[ FAIL ] " << c.name << ": " << e.message << "\n";
75	            ++failures;
76	        } catch (const std::exception& e) {
77	            std::cout << "[ FAIL ] " << c.name
78	                      << ": unexpected exception: " << e.what() << "\n";
79	            ++failures;
80	        } catch (...) {
81	            std::cout << "[ FAIL ] " << c.name << ": unknown exception\n";
82	            ++failures;
83	        }
84	    }
85	    std::cout << "----\n"
86	              << (registry().size() - failures) << "/" << registry().size()
87	              << " tests passed\n";
88	    return failures == 0 ? 0 : 1;
89	}
90	
91	} // namespace beamfd_test
92	
93	#define BEAMFD_CONCAT_INNER(a, b) a##b
94	#define BEAMFD_CONCAT(a, b) BEAMFD_CONCAT_INNER(a, b)
95	
96	#define BEAMFD_TEST(NAME)                                                      \
97	    static void BEAMFD_CONCAT(beamfd_test_fn_, __LINE__)();                    \
98	    static ::beamfd_test::Registrar BEAMFD_CONCAT(beamfd_test_reg_,           \
99	                                                  __LINE__){                   \
100	        NAME, &BEAMFD_CONCAT(beamfd_test_fn_, __LINE__)};                      \
101	    static void BEAMFD_CONCAT(beamfd_test_fn_, __LINE__)()
102	
103	#define BEAMFD_TEST_MAIN()                                                     \
104	    int main() { return ::beamfd_test::run_all(); }
105	
106	#endif // BEAMFD_TEST_UTIL_HPP
107
This is a clear, well-specified numerical task. Let me look at the io/demo and CMake to understand the build, then implement.

/opt/beamfd/tests/CMakeLists.txt

contents
1	# The shipped project ships only a tiny, non-leaky smoke test using the
2	# header-only harness in test_util.hpp. The authoritative grading is a separate
3	# HIDDEN suite injected by the verifier (tests/test.sh) at grade time, so do
4	# not rely on this smoke test to characterize correctness.
5	
6	add_executable(test_smoke test_smoke.cpp)
7	target_link_libraries(test_smoke PRIVATE beamfd)
8	target_include_directories(test_smoke PRIVATE ${CMAKE_CURRENT_SOURCE_DIR})
9	add_test(NAME smoke COMMAND test_smoke)
10

List project tree excluding build

shell
ls -R /opt/beamfd --ignore=build 2>/dev/null | head -60
output
/opt/beamfd:
CMakeLists.txt
examples
include
src
tests

/opt/beamfd/examples:
cantilever_tip.beam

/opt/beamfd/include:
beamfd

/opt/beamfd/include/beamfd:
beam.hpp
io.hpp
linalg.hpp
solver.hpp
types.hpp

/opt/beamfd/src:
beam.cpp
io.cpp
linalg.cpp
main.cpp
solver.cpp

/opt/beamfd/tests:
CMakeLists.txt
test_smoke.cpp
test_util.hpp

[stdout]
/opt/beamfd:
CMakeLists.txt
examples
include
src
tests

/opt/beamfd/examples:
cantilever_tip.beam

/opt/beamfd/include:
beamfd

/opt/beamfd/include/beamfd:
beam.hpp
io.hpp
linalg.hpp
solver.hpp
types.hpp

/opt/beamfd/src:
beam.cpp
io.cpp
linalg.cpp
main.cpp
solver.cpp

/opt/beamfd/tests:
CMakeLists.txt
test_smoke.cpp
test_util.hpp
I now have a complete understanding. Let me implement the finite-difference core. My scheme: - **Interior PDE** (rows 1..N-1): the operator `(EI w'')''` discretized as the second central difference of `EI·w''` (a double second-difference), giving the standard `[1,-4,6,-4,1]·EI/dx⁴` stencil for constant EI, plus `+k_i w_i` and RHS `q_i`. - **Ghost elimination** at the near-boundary rows 1 and N-1 using the *secondary* BC at each end (slope for clamped, moment for pinned/free) , all second-order central differences. - **End rows** 0 and N hold the *primary* condition: `w=0` for clamped/pinned, or a second-order one-sided shear condition for free ends.

/opt/beamfd/src/solver.cpp

− removed
BeamSystem BeamSolver::assemble_system() const {
    // TODO(candidate): build and return the finite-difference system A w = b.
    const std::size_t n = beam_.num_nodes();
    BeamSystem sys;
    sys.A = DenseMatrix(n, n);  // all zeros -> singular / wrong
    sys.b.assign(n, 0.0);
    return sys;
}

BeamResult BeamSolver::solve() const {
    // TODO(candidate): validate, assemble, solve, return the deflection field.
    const std::size_t n = beam_.num_nodes();
    BeamResult result;
    result.deflection.assign(n, 0.0);  // trivially zero -> wrong
    return result;
}
+ added
BeamSystem BeamSolver::assemble_system() const {
    // Discretize  d2/dx2( EI(x) w''(x) ) + k(x) w(x) = q(x)  on the uniform grid
    // x_i = i*dx, i = 0..N, with n = N+1 nodal unknowns w_i.
    //
    // Interior operator (second-order). The fourth-order term is the second
    // central difference of the nodal moment M_j = EI_j * w''_j:
    //
    //   (EI w'')''_i ~= ( M_{i-1} - 2 M_i + M_{i+1} ) / dx^2 ,
    //   M_j         ~= EI_j ( w_{j-1} - 2 w_j + w_{j+1} ) / dx^2 ,
    //
    // which expands to the 5-point stencil (coefficients divided by dx^4)
    //
    //   EI_{i-1} w_{i-2}
    //   - 2(EI_{i-1}+EI_i) w_{i-1}
    //   + (EI_{i-1}+4 EI_i+EI_{i+1}) w_i
    //   - 2(EI_i+EI_{i+1}) w_{i+1}
    //   + EI_{i+1} w_{i+2}
    //
    // and reduces to the classic EI/dx^4 * [1,-4,6,-4,1] biharmonic stencil for
    // constant EI. The PDE is collocated at the interior nodes i = 1..N-1; the
    // single ghost each such near-boundary row reaches (w_{-1} at row 1, w_{N+1}
    // at row N-1) is removed using the end's secondary boundary condition, all
    // second-order accurate. Rows 0 and N hold the end's primary condition.
    const std::size_t n = beam_.num_nodes();
    const std::size_t N = beam_.num_intervals();
    const double h = beam_.dx();
    const double h2 = h * h;
    const double h4 = h2 * h2;

    BeamSystem sys;
    sys.A = DenseMatrix(n, n);
    sys.b.assign(n, 0.0);
    DenseMatrix& A = sys.A;

    // Add to A(r, c), silently ignoring out-of-range columns. Out-of-range only
    // happens for under-resolved meshes (N < 4) where the 4th-order stencil
    // cannot fit; for any adequately refined grid every term lands in range.
    auto add = [&](std::size_t r, long c, double v) {
        if (c >= 0 && static_cast<std::size_t>(c) < n) {
            A(r, static_cast<std::size_t>(c)) += v;
        }
    };

    // ---- Interior PDE rows i = 1 .. N-1 -------------------------------------
    for (std::size_t i = 1; i + 1 < n; ++i) {
        const double eim = beam_.EI_at(i - 1);
        const double eii = beam_.EI_at(i);
        const double eip = beam_.EI_at(i + 1);
        const long li = static_cast<long>(i);
        add(i, li - 2, eim / h4);
        add(i, li - 1, -2.0 * (eim + eii) / h4);
        add(i, li,    (eim + 4.0 * eii + eip) / h4);
        add(i, li + 1, -2.0 * (eii + eip) / h4);
        add(i, li + 2, eip / h4);
        add(i, li, beam_.k_at(i));  // Winkler foundation
        sys.b[i] = beam_.q_at(i);
    }

    // ---- Ghost elimination at the two near-boundary interior rows ----------
    // Each end's secondary condition expresses the exterior ghost as a linear
    // combination of the two nearest real nodes plus a constant:
    //   left : w_{-1}  = a0 w_0     + a1 w_1     + cL
    //   right: w_{N+1} = b0 w_N     + b1 w_{N-1} + cR
    //   clamped (w'=0):        a0=0, a1= 1, c=0
    //   pinned  (w''=0):       a0=2, a1=-1, c=0
    //   free    (EI w''=M):    a0=2, a1=-1, c = M_applied * dx^2 / EI_end
    if (n >= 3) {
        // Left ghost folded into PDE row 1 (coefficient on w_{-1} is EI_0/dx^4).
        const double cL = beam_.EI_at(0) / h4;
        double a0, a1, constL = 0.0;
        switch (beam_.left()) {
            case Support::Clamped: a0 = 0.0; a1 = 1.0; break;
            case Support::Pinned:  a0 = 2.0; a1 = -1.0; break;
            case Support::Free:
                a0 = 2.0; a1 = -1.0;
                constL = beam_.end_moment(true) * h2 / beam_.EI_at(0);
                break;
        }
        add(1, 0, cL * a0);
        add(1, 1, cL * a1);
        sys.b[1] -= cL * constL;

        // Right ghost folded into PDE row N-1 (coefficient on w_{N+1} is
        // EI_N/dx^4).
        const double cR = beam_.EI_at(N) / h4;
        double b0, b1, constR = 0.0;
        switch (beam_.right()) {
            case Support::Clamped: b0 = 0.0; b1 = 1.0; break;
            case Support::Pinned:  b0 = 2.0; b1 = -1.0; break;
            case Support::Free:
                b0 = 2.0; b1 = -1.0;
                constR = beam_.end_moment(false) * h2 / beam_.EI_at(N);
                break;
        }
        add(N - 1, static_cast<long>(N), cR * b0);
        add(N - 1, static_cast<long>(N) - 1, cR * b1);
        sys.b[N - 1] -= cR * constR;
    }

    // ---- Primary boundary rows 0 and N -------------------------------------
    // Clamped / pinned: essential condition w = 0.
    // Free: the shear V = (EI w'')' equals the applied end shear. Using a
    // second-order one-sided difference of the nodal moment, with M at the end
    // node fixed to the applied moment by the ghost relation above:
    //   V_0 = (-3 M_0 + 4 M_1 - M_2)/(2 dx) = V_app, M_0 = M_app
    //     => 4 M_1 - M_2 = 2 dx V_app + 3 M_app
    //   V_N = ( 3 M_N - 4 M_{N-1} + M_{N-2})/(2 dx) = V_app, M_N = M_app
    //     => -4 M_{N-1} + M_{N-2} = 2 dx V_app - 3 M_app
    // with M_j = EI_j (w_{j-1} - 2 w_j + w_{j+1})/dx^2.
    if (beam_.left() == Support::Free) {
        const double ei1 = beam_.EI_at(1);
        const double ei2 = beam_.EI_at(2);
        // 4 M_1 - M_2, coefficients divided by dx^2.
        add(0, 0, 4.0 * ei1 / h2);
        add(0, 1, (-8.0 * ei1 - ei2) / h2);
        add(0, 2, (4.0 * ei1 + 2.0 * ei2) / h2);
        add(0, 3, -ei2 / h2);
        sys.b[0] = 2.0 * h * beam_.applied_end_shear(true) +
                   3.0 * beam_.end_moment(true);
    } else {
        A(0, 0) = 1.0;
        sys.b[0] = 0.0;
    }

    if (beam_.right() == Support::Free) {
        const double eim1 = beam_.EI_at(N - 1);
        const double eim2 = beam_.EI_at(N - 2);
        // -4 M_{N-1} + M_{N-2}, coefficients divided by dx^2.
        add(N, static_cast<long>(N), -4.0 * eim1 / h2);
        add(N, static_cast<long>(N) - 1, (8.0 * eim1 + eim2) / h2);
        add(N, static_cast<long>(N) - 2, (-4.0 * eim1 - 2.0 * eim2) / h2);
        add(N, static_cast<long>(N) - 3, eim2 / h2);
        sys.b[N] = 2.0 * h * beam_.applied_end_shear(false) -
                   3.0 * beam_.end_moment(false);
    } else {
        A(N, N) = 1.0;
        sys.b[N] = 0.0;
    }

    return sys;
}

BeamResult BeamSolver::solve() const {
    beam_.validate();  // propagates mechanism / configuration errors
    const BeamSystem sys = assemble_system();
    BeamResult result;
    // solve_general throws std::runtime_error on a singular system; let it
    // propagate per the contract.
    result.deflection = solve_general(sys.A, sys.b);
    return result;
}
Now let me build and run the smoke test.

Configure, build, run smoke test

shell
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release >/dev/null 2>&1 && cmake --build build --parallel 2>&1 | tail -15 && cd build && ctest --output-on-failure 2>&1 | tail -20
output
|         ~~~^~~~~~~~~~~~~~~
/opt/beamfd/src/solver.cpp:133:12: warning: 'b0' may be used uninitialized [-Wmaybe-uninitialized]
  133 |         add(N - 1, static_cast<long>(N), cR * b0);
      |         ~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
/opt/beamfd/src/solver.cpp:134:12: warning: 'b1' may be used uninitialized [-Wmaybe-uninitialized]
  134 |         add(N - 1, static_cast<long>(N) - 1, cR * b1);
      |         ~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[ 55%] Linking CXX static library libbeamfd.a
[ 55%] Built target beamfd
[ 66%] Building CXX object CMakeFiles/beamfd_cli.dir/src/main.cpp.o
[ 77%] Building CXX object tests/CMakeFiles/test_smoke.dir/test_smoke.cpp.o
[ 88%] Linking CXX executable beamfd_cli
[ 88%] Built target beamfd_cli
[100%] Linking CXX executable test_smoke
[100%] Built target test_smoke
Test project /opt/beamfd/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.01 sec

[stdout]
      |         ~~~^~~~~~~~~~~~~~~
/opt/beamfd/src/solver.cpp:133:12: warning: 'b0' may be used uninitialized [-Wmaybe-uninitialized]
  133 |         add(N - 1, static_cast<long>(N), cR * b0);
      |         ~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
/opt/beamfd/src/solver.cpp:134:12: warning: 'b1' may be used uninitialized [-Wmaybe-uninitialized]
  134 |         add(N - 1, static_cast<long>(N) - 1, cR * b1);
      |         ~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[ 55%] Linking CXX static library libbeamfd.a
[ 55%] Built target beamfd
[ 66%] Building CXX object CMakeFiles/beamfd_cli.dir/src/main.cpp.o
[ 77%] Building CXX object tests/CMakeFiles/test_smoke.dir/test_smoke.cpp.o
[ 88%] Linking CXX executable beamfd_cli
[ 88%] Built target beamfd_cli
[100%] Linking CXX executable test_smoke
[100%] Built target test_smoke
Test project /opt/beamfd/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.01 sec
Smoke passes. Let me silence the uninitialized warnings (switch is exhaustive but the compiler can't tell), then validate accuracy against analytical solutions.

/opt/beamfd/src/solver.cpp

− removed
        const double cL = beam_.EI_at(0) / h4;
        double a0, a1, constL = 0.0;
+ added
        const double cL = beam_.EI_at(0) / h4;
        double a0 = 0.0, a1 = 1.0, constL = 0.0;

/opt/beamfd/src/solver.cpp

− removed
        const double cR = beam_.EI_at(N) / h4;
        double b0, b1, constR = 0.0;
+ added
        const double cR = beam_.EI_at(N) / h4;
        double b0 = 0.0, b1 = 1.0, constR = 0.0;
Now let me write a validation program testing several analytical cases and convergence order.
write · /opt/beamfd/tests/validate.cpp
// Local validation against closed-form solutions + convergence order.
#include <cmath>
#include <cstdio>
#include <vector>
#include "beamfd/beam.hpp"
#include "beamfd/solver.hpp"
using namespace beamfd;

static double max_rel_err(const std::vector<double>& w,
                          const std::vector<double>& exact) {
    double num = 0, den = 0;
    for (size_t i = 0; i < w.size(); ++i) {
        num = std::max(num, std::fabs(w[i] - exact[i]));
        den = std::max(den, std::fabs(exact[i]));
    }
    return den > 0 ? num / den : num;
}

// returns max-norm absolute error
template <class F>
static double run(Support l, Support r, int N, double L, double EI, F exact,
                  double q = 0, double Pend = 0, bool Pleft = false,
                  double Mend = 0, bool Mleft = false) {
    Beam beam(L, EI, N, l, r);
    if (q != 0) beam.set_distributed_load(q);
    if (Pend != 0) beam.set_end_load(Pend, Pleft);
    if (Mend != 0) beam.set_end_moment(Mend, Mleft);
    BeamSolver s(beam);
    auto w = s.solve().deflection;
    std::vector<double> ex(w.size());
    double e = 0;
    for (size_t i = 0; i < w.size(); ++i) {
        ex[i] = exact(beam.node_x(i));
        e = std::max(e, std::fabs(w[i] - ex[i]));
    }
    return e;
}

static void order(const char* name, double e1, double e2) {
    double p = std::log2(e1 / e2);
    printf("%-40s err(N)=%.3e err(2N)=%.3e  order~%.2f\n", name, e1, e2, p);
}

int main() {
    const double L = 2.0, EI = 1234.0;

    // 1. Simply supported (pinned-pinned), UDL q. w = q x(L^3-2Lx^2+x^3)/(24EI)
    {
        double q = 7.0;
        auto ex = [&](double x){ return q*x*(L*L*L - 2*L*x*x + x*x*x)/(24*EI); };
        double e1 = run(Support::Pinned, Support::Pinned, 40, L, EI, ex, q);
        double e2 = run(Support::Pinned, Support::Pinned, 80, L, EI, ex, q);
        order("pinned-pinned UDL", e1, e2);
    }
    // 2. Cantilever clamped-left free-right, UDL. w=q x^2(x^2-4Lx+6L^2)/(24EI)
    {
        double q = 5.0;
        auto ex = [&](double x){ return q*x*x*(x*x - 4*L*x + 6*L*L)/(24*EI); };
        double e1 = run(Support::Clamped, Support::Free, 40, L, EI, ex, q);
        double e2 = run(Support::Clamped, Support::Free, 80, L, EI, ex, q);
        order("cantilever UDL", e1, e2);
    }
    // 3. Cantilever, tip load P (downward, at right). w = P x^2(3L-x)/(6EI)
    {
        double P = 9.0;
        auto ex = [&](double x){ return P*x*x*(3*L - x)/(6*EI); };
        double e1 = run(Support::Clamped, Support::Free, 40, L, EI, ex, 0, P, false);
        double e2 = run(Support::Clamped, Support::Free, 80, L, EI, ex, 0, P, false);
        order("cantilever tip load", e1, e2);
    }
    // 4. Cantilever, tip moment M at right. w = M x^2/(2EI)
    {
        double M = 11.0;
        auto ex = [&](double x){ return M*x*x/(2*EI); };
        double e1 = run(Support::Clamped, Support::Free, 40, L, EI, ex, 0,0,false, M, false);
        double e2 = run(Support::Clamped, Support::Free, 80, L, EI, ex, 0,0,false, M, false);
        order("cantilever tip moment", e1, e2);
    }
    // 5. Clamped-clamped UDL. w = q x^2 (L-x)^2 /(24 EI)
    {
        double q = 4.0;
        auto ex = [&](double x){ return q*x*x*(L-x)*(L-x)/(24*EI); };
        double e1 = run(Support::Clamped, Support::Clamped, 40, L, EI, ex, q);
        double e2 = run(Support::Clamped, Support::Clamped, 80, L, EI, ex, q);
        order("clamped-clamped UDL", e1, e2);
    }
    // 6. Clamped-pinned UDL. w = q x^2 (3L^2 -5Lx +2x^2)/(48 EI)
    {
        double q = 6.0;
        auto ex = [&](double x){ return q*x*x*(3*L*L -5*L*x +2*x*x)/(48*EI); };
        double e1 = run(Support::Clamped, Support::Pinned, 40, L, EI, ex, q);
        double e2 = run(Support::Clamped, Support::Pinned, 80, L, EI, ex, q);
        order("clamped-pinned UDL", e1, e2);
    }
    // 7. Manufactured variable EI + foundation, free-free with foundation:
    //    pick w(x), EI(x), k(x), compute q = (EI w'')'' + k w, set end actions.
    {
        // w = sin(pi x / L); EI = EI0(1+0.5 x/L); k = k0(1+ x/L)
        double EI0 = 800.0, k0 = 50.0, kc = M_PI/L;
        auto W   = [&](double x){ return std::sin(kc*x); };
        auto W2  = [&](double x){ return -kc*kc*std::sin(kc*x); };
        auto W3  = [&](double x){ return -kc*kc*kc*std::cos(kc*x); };
        auto W4  = [&](double x){ return kc*kc*kc*kc*std::sin(kc*x); };
        auto E   = [&](double x){ return EI0*(1+0.5*x/L); };
        auto Ep  = [&](double x){ return EI0*0.5/L; };
        auto K   = [&](double x){ return k0*(1+x/L); };
        // (EI w'')'' = EI'' w'' + 2 EI' w''' + EI w''''; EI''=0
        auto Q   = [&](double x){ return 2*Ep(x)*W3(x) + E(x)*W4(x) + K(x)*W(x); };
        // M = EI w''; V = (EI w'')' = EI' w'' + EI w'''
        auto Vf  = [&](double x){ return Ep(x)*W2(x) + E(x)*W3(x); };
        auto Mf  = [&](double x){ return E(x)*W2(x); };

        auto build = [&](int N){
            Beam beam(L, EI0, N, Support::Free, Support::Free);
            std::vector<double> ei(N+1), kk(N+1), qq(N+1);
            for (int i=0;i<=N;i++){ double x=beam.node_x(i); ei[i]=E(x); kk[i]=K(x); qq[i]=Q(x);}
            beam.set_ei_nodal(ei); beam.set_foundation_nodal(kk); beam.set_q_nodal(qq);
            // end actions: applied_end_shear = V required; but setter takes force P with V=-P
            // left: V(0) = Vf(0) => P_left = -Vf(0)
            beam.set_end_load(-Vf(0.0), true);
            beam.set_end_load( Vf(L),  false);
            beam.set_end_moment(Mf(0.0), true);
            beam.set_end_moment(Mf(L),  false);
            BeamSolver s(beam);
            auto w=s.solve().deflection; double e=0;
            for(int i=0;i<=N;i++) e=std::max(e,std::fabs(w[i]-W(beam.node_x(i))));
            return e;
        };
        double e1=build(40), e2=build(80);
        order("varEI+foundation free-free (mfd)", e1, e2);
    }
    return 0;
}

/opt/beamfd/tests/validate.cpp

− removed
            beam.set_end_load(-Vf(0.0), true);
            beam.set_end_load( Vf(L),  false);
+ added
            beam.set_end_load(-Vf(0.0), true);
            beam.set_end_load(-Vf(L),  false);

Compile and run validation

shell
cd /opt/beamfd && g++ -std=c++17 -O2 -Iinclude tests/validate.cpp src/beam.cpp src/solver.cpp src/linalg.cpp -o /tmp/validate 2>&1 | tail -20 && /tmp/validate
output
pinned-pinned UDL                        err(N)=5.909e-07 err(2N)=1.477e-07  order~2.00
cantilever UDL                           err(N)=5.065e-06 err(2N)=1.266e-06  order~2.00
cantilever tip load                      err(N)=6.078e-06 err(2N)=1.519e-06  order~2.00
cantilever tip moment                    err(N)=1.253e-12 err(2N)=1.731e-11  order~-3.79
clamped-clamped UDL                      err(N)=6.753e-07 err(2N)=1.688e-07  order~2.00
clamped-pinned UDL                       err(N)=1.084e-06 err(2N)=2.711e-07  order~2.00
varEI+foundation free-free (mfd)         err(N)=1.692e-01 err(2N)=4.340e-02  order~1.96

[stdout]
pinned-pinned UDL                        err(N)=5.909e-07 err(2N)=1.477e-07  order~2.00
cantilever UDL                           err(N)=5.065e-06 err(2N)=1.266e-06  order~2.00
cantilever tip load                      err(N)=6.078e-06 err(2N)=1.519e-06  order~2.00
cantilever tip moment                    err(N)=1.253e-12 err(2N)=1.731e-11  order~-3.79
clamped-clamped UDL                      err(N)=6.753e-07 err(2N)=1.688e-07  order~2.00
clamped-pinned UDL                       err(N)=1.084e-06 err(2N)=2.711e-07  order~2.00
varEI+foundation free-free (mfd)         err(N)=1.692e-01 err(2N)=4.340e-02  order~1.96
All cases are **second-order accurate** (order ~2.00). The "cantilever tip moment" case shows machine-precision error (1e-12) because its exact solution is quadratic, which the scheme reproduces *exactly* , confirming the free-end moment/shear handling is correct. The variable-EI + foundation + free-free manufactured solution converges at order ~1.96. Let me clean up the temporary validation file and do a final build verification.

Remove temp file, rebuild, run tests

shell
rm -f tests/validate.cpp && cmake --build build --parallel 2>&1 | grep -iE "error|warning" | head; echo "---build clean---"; cd build && ctest --output-on-failure 2>&1 | tail -5
output
---build clean---
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.00 sec

[stdout]
---build clean---
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.00 sec
The implementation is complete, builds cleanly (no warnings), and the smoke test passes. ## Summary I implemented the two stubbed functions in `/opt/beamfd/src/solver.cpp`. Nothing else was changed , public headers, signatures, recovery helpers, and the support layer are all untouched. ### Discretization scheme (second-order, conservative form) The governing equation `d²/dx²(EI(x)·w'') + k(x)·w = q(x)` is discretized on the uniform grid as a **double second-difference**: `(EI w'')''ᵢ ≈ (Mᵢ₋₁ − 2Mᵢ + Mᵢ₊₁)/dx²` with `Mⱼ = EIⱼ(wⱼ₋₁ − 2wⱼ + wⱼ₊₁)/dx²`. This expands to a 5-point stencil that correctly honors variable `EI(x)` and reduces to the classic `EI/dx⁴·[1,−4,6,−4,1]` biharmonic stencil for constant `EI`. The Winkler term `+kᵢwᵢ` goes on the diagonal; `qᵢ` is the RHS. The PDE is collocated at interior nodes `1..N−1`. ### Boundary conditions (all second-order) Each end contributes two conditions. The **secondary** condition eliminates the one exterior ghost node that the near-boundary PDE row reaches (via central differences): - **Clamped**: `w'=0` → `w₋₁ = w₁` (yields the well-known "7" near-boundary coefficient) - **Pinned**: `w''=0` → `w₋₁ = 2w₀ − w₁` (yields the "5" coefficient) - **Free**: `EI w'' = M_applied` → defines the ghost and fixes the end moment exactly The **primary** condition is the end row: `w=0` for clamped/pinned (essential), or a one-sided second-order shear condition `(EI w'')' = V_applied` for free ends, using `applied_end_shear`/`end_moment` directly so sign conventions are delegated to `Beam`. `solve()` calls `Beam::validate()` first and lets its exceptions , plus any singular-system exception from `solve_general` , propagate, per the contract. ### Verification I validated against seven independently manufactured solutions (pinned-pinned, cantilever under UDL / tip load / tip moment, clamped-clamped, clamped-pinned, and a variable-`EI` + foundation + free-free case with applied end actions). Every case showed **clean second-order mesh convergence (order ≈ 2.00)**; the tip-moment case is reproduced to machine precision since its exact solution is quadratic.
Final diff , what the agent changed

No file changes, the agent left the workspace untouched.

Verifier output , the proof behind the reward
tests/test.sh
== Configuring (cmake) ==
-- The CXX compiler identification is GNU 11.4.0
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Check for working CXX compiler: /usr/bin/c++ - skipped
-- Detecting CXX compile features
-- Detecting CXX compile features - done
-- Configuring done
-- Generating done
-- Build files have been written to: /tmp/tmp.bnfEtutN9k/beamfd/build_grade
== Building ==
[  6%] Building CXX object CMakeFiles/beamfd.dir/src/io.cpp.o
[ 13%] Building CXX object CMakeFiles/beamfd.dir/src/linalg.cpp.o
[ 20%] Building CXX object CMakeFiles/beamfd.dir/src/beam.cpp.o
[ 26%] Building CXX object CMakeFiles/beamfd.dir/src/solver.cpp.o
[ 33%] Linking CXX static library libbeamfd.a
[ 33%] Built target beamfd
[ 40%] Building CXX object CMakeFiles/beamfd_cli.dir/src/main.cpp.o
[ 53%] Building CXX object tests/CMakeFiles/test_mms.dir/test_mms.cpp.o
[ 53%] Building CXX object tests/CMakeFiles/test_invariants.dir/test_invariants.cpp.o
[ 60%] Building CXX object tests/CMakeFiles/test_convergence.dir/test_convergence.cpp.o
[ 66%] Building CXX object tests/CMakeFiles/test_soak.dir/test_soak.cpp.o
[ 73%] Linking CXX executable beamfd_cli
[ 73%] Built target beamfd_cli
[ 80%] Linking CXX executable test_convergence
[ 86%] Linking CXX executable test_mms
[ 86%] Built target test_convergence
[ 93%] Linking CXX executable test_soak
[100%] Linking CXX executable test_invariants
[100%] Built target test_soak
[100%] Built target test_mms
[100%] Built target test_invariants
== Running hidden test suite (ctest) ==
Test project /tmp/tmp.bnfEtutN9k/beamfd/build_grade
    Start 1: test_mms
1/4 Test #1: test_mms .........................   Passed    0.01 sec
    Start 2: test_convergence
2/4 Test #2: test_convergence .................   Passed    0.01 sec
    Start 3: test_invariants
3/4 Test #3: test_invariants ..................   Passed    0.00 sec
    Start 4: test_soak
4/4 Test #4: test_soak ........................   Passed    0.01 sec

100% tests passed, 0 tests failed out of 4

Label Time Summary:
hidden    =   0.03 sec*proc (4 tests)

Total Test time (real) =   0.03 sec
PASS: all hidden tests passed

Reproduce this trial: git checkout 2f94510 && PYTHONPATH=src python3 scripts/build_site.py , then open trial/trial_e56e79f512e3436e. Re-running the agent live requires EVAL_PLATFORM_ENABLE_OAUTH_SMOKE=1 and is non-deterministic.

Trial trial_e56e79f512e3436e · verifier authoritative; classifier explanatory.