{"submission_id":"c07160ef-27e4-420e-b70e-0326f26d4351","submission_revision_id":"47fa14c9-ab22-4f0e-b50e-d92e3aa84da7","revision_version":2,"submission_status":"submitted","submitted_at":"2026-08-17T02:28:29.440238Z","submitter_name":"Haoxuan Chen","submitter_email":"haoxuan@alumni.caltech.edu","task":{"status":"submitted","version":2,"extra_fields":{},"submitted_at":"2026-08-17T02:28:29.440238+00:00","industry_domain":"Numerical Analysis and Scientific Computing","task_short_name":"High-order FEM methods for PDEs","operating_system":"Linux","task_description":"You are working on a Linux coding task about `C¹` finite element methods for a\nfourth-order two-point boundary value problem.\n\n**Task directory:** `base`\n\n**Visible input files:**\n- Problem specification: `base/input/problem_spec.md`\n- Python runtime manifest: `base/input/runtime_env/pyproject.toml`\n- Python lockfile: `base/input/runtime_env/uv.lock`\n- Canonical Python wrapper: `base/software/python.sh`\n\n**Recommended workflow:**\n1. Change into `base`.\n2. Read `base/input/problem_spec.md` in full before coding.\n3. Use `\"base/software/python.sh\" your_script.py` or\n   `\"base/software/python.sh\" -c \"...\"` to run Python with the staged\n   NumPy/SciPy environment.\n4. Implement your solver and save it to `base/output/fem_solver.py`.\n5. Run the solver and write every required artifact under `base/output`.\n\n**Required outputs under `base/output`:** `fem_solver.py`,\n`reference_arrays.npz`, `results.json`.\n\n**Objective:** Implement a **C¹ cubic-Hermite finite element method** for the\nclamped Euler–Bernoulli beam — a fourth-order BVP `u'''' = f` on `(0,1)` with\n`u(0)=u'(0)=u(1)=u'(1)=0` — and demonstrate fourth-order convergence against a\nmanufactured solution. Because the weak form `(u'', v'') = (f, v)` pairs *second*\nderivatives, the method needs `C¹` elements (value **and** slope DOFs), not\nstandard `C⁰` Lagrange. Manufactured `u_e = eˣ·x²(1−x)²`,\n`f = eˣ(x⁴+14x³+49x²+32x−12)`.\n\n**Instructions & Rules:**\n1. Implement the solver in `base/output/fem_solver.py` using **Python + NumPy +\n   SciPy only** — no FEM/PDE library; derive the element stiffness, assemble the\n   `2(n−1)` system under the clamped BCs, build the load, and solve.\n2. Integrate the load with a rule that is at least **3-point Gauss–Legendre per\n   element** (or an exact integral). A cruder rule destroys the nodal\n   superconvergence that step 4 asks you to exhibit.\n3. `fem_solver.py` must be importable without side effects and expose\n   `solve(n) -> (x_nodes, u_nodes, x_dense, u_dense)` with shapes `(n+1,)`,\n   `(n+1,)`, `(10n+1,)`, `(10n+1,)`, on the grids `x_nodes[j] = j/n` and\n   `x_dense[i] = i/(10n)`. `u_dense` is the FE field reconstructed from **all\n   four** local DOFs (values *and* slopes).\n4. **Two error measures (core):** for `n = 2^p`, `p = 5..11`, report **both** the\n   nodal error `e_n = max_j |u_e(x_j) − u_h(x_j)|` and the dense (10×-finer)\n   error `e*_n = max_i |u_e(y_i) − u_h(y_i)|`, `y_i = i/(10n)`, with the orders\n   `β = log₂(e_n/e_{2n})`. Determine the asymptotic order and explain how the two\n   measures differ as `n` grows.\n5. **Artifact:** at `n = 256`, save the dense reconstruction of `u_h` over the\n   2561 points `y_i = i/2560`, the exact solution, and the pointwise error.\n   `max|uh_dense − uexact_dense|` must equal the `n = 256` row of the dense\n   study — the artifact and the table must come from the same run.\n6. Write `reference_arrays.npz` and a consistent `results.json`; keep every saved\n   value truthful (the grader recomputes the exact solution and all orders, and\n   re-runs your solver).\n7. Do not modify files outside `base/output`.","software_licensing":"Free/Open Source","evaluation_criteria":null,"verification_method":"An automated grader (`evaluation/grade.py`, provided) applies **strict, objective**\nrules. It knows the manufactured exact solution, recomputes it analytically,\nrecomputes every order of convergence from the submitted arrays, **and re-runs\nthe submitted solver in an isolated subprocess** — so a non-conforming element, a\nmis-scaled slope DOF, an under-integrated load, a slope-omitting reconstruction,\nor fabricated metrics all fail.\n\n**Scoring**\n- `1.0` — Tier 1 + Tier 2 + Tier 3 all pass.\n- `0.5` — Tier 1 + Tier 2 and all provenance checks pass, Tier 3 missing/fails.\n- `0.0` — required files missing (including `fem_solver.py`), array schema\n  violations, `results.json` inconsistent with the arrays, wrong mesh sequence,\n  non-finite or non-positive errors, a disallowed dependency, a solver that\n  cannot reproduce its own submitted numbers or returns a nonconforming grid, or\n  Tier 1 / Tier 2 fail.\n\n**Objective thresholds** (exact reference magnitudes live only in the uploaded\n`grade.py` / `EVALUATION.md`, which are evaluation dependencies — not part of the\nagent's task input). The specification fixes the method completely, so a band\naround the reference dense errors is a statement about the problem, not a\npreference among admissible schemes:\n- Tier 1: dense `e*_n` at `n = 32, 64, 128` within ±25 % of the gold reference;\n  the two coarse dense orders in `[3.8, 4.2]`.\n- Tier 2: ≥ 2 of the three clean-regime dense orders (`n = 32→256`) in\n  `[3.8, 4.2]`; nodal `e_n(32) < 1e-8` and `< e*_n(32)/100` (superconvergence);\n  and a round-off floor (dense error at `n = 2048` exceeds `10×` that at\n  `n = 256`).\n- Tier 3: `x_dense` equal to the exact grid `i/2560` (to `1e-15`), running from\n  exactly `0.0` to `1.0` and strictly increasing; `uexact_dense` equal to the\n  analytic solution; recomputed max-norm error `< 1e-9`;\n  `error_dense == uh_dense − uexact_dense`; and that recomputed error equal to\n  `dense_errors[3]`, proving the artifact and the table's `n = 256` row come from\n  the same run.\n- Provenance: `fem_solver.py` present, importable, exposing `solve(n)`, importing\n  nothing outside stdlib + NumPy + SciPy, returning the prescribed grids, and\n  regenerating both error measures to within `1e-6` relative of the submitted\n  values. **All seven meshes are regenerated and must match** — nothing is\n  best-effort, and a timeout is a failure rather than an accepted gap. The\n  `n = 256` run is the artifact run, so the submitted dense field is also compared pointwise against the freshly generated one.\n- Scheme fidelity: the grader carries its own cubic-Hermite implementation and\n  compares the submitted dense field against it pointwise at **every** mesh\n  `n = 32 … 2048` (`1e-9`). It additionally observes a global solve of the\n  required `2(n−1)` free-DOF system at every mesh and, at `n = 32`, poisons solve\n  results to verify that the returned field causally depends on them. This closes\n  the exact-Hermite-interpolant and discarded-dummy-solve loopholes; an\n  unobservable hand-written solve is flagged for manual review.\n- Consistency: `results.json` must agree with `reference_arrays.npz` on both study\n  tables, the `h` values, every `beta`, the claimed orders, and the artifact\n  metadata.\n\nThe discriminating check is the **dense (between-node) error**: a reconstruction\nthat omits the Hermite slope DOFs still produces *perfect nodal values* — its\nnodal study is numerically identical to the gold one — but collapses the dense\nerror to order 1 (measured: `1.16e-03` at `n = 32` against `5.40e-07`, orders\n`1.01, 1.00, 1.00`). `EVALUATION.md` documents this calibration, along with the\nplaces where an automated check is inconclusive and degrades to a manual-review note rather than a false failure.","software_and_version":"Python >=3.11,<3.14, NumPy 2.4.6, SciPy 1.17.1","specific_requirements":null,"expected_completion_time":null},"files":[{"file_id":"77dc6b71-7be6-4040-8778-8d6531f56b5c","filename":"problem_spec.md","category":"input_materials","content_type":"text/markdown","size_bytes":10748},{"file_id":"3b8e363e-0cc4-4323-81d1-cf57fa920f6b","filename":"python.sh","category":"input_materials","content_type":"application/x-sh","size_bytes":2143},{"file_id":"efe0ff68-9526-4a08-bec4-793d7d5e7fe3","filename":"pyproject.toml","category":"input_materials","content_type":"application/octet-stream","size_bytes":264},{"file_id":"327dd4ac-a008-466c-b07b-933cf2d11aa6","filename":"uv.lock","category":"input_materials","content_type":"application/octet-stream","size_bytes":29647},{"file_id":"719dc09b-58bb-4760-81ef-0929c2c5771b","filename":"grade.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":53426},{"file_id":"84786a2f-e0e5-415c-9a6f-e62c532a34d2","filename":"EVALUATION.md","category":"reference_output","content_type":"text/markdown","size_bytes":11943},{"file_id":"488f743a-94f6-458f-bef8-733840134694","filename":"fem_solver.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":9185},{"file_id":"00d5ea87-3b98-44cf-a911-d321d665428c","filename":"results.json","category":"reference_output","content_type":"application/json","size_bytes":2735},{"file_id":"9b16723b-788d-4d1d-9a7c-c403ab661861","filename":"reference_arrays.npz","category":"reference_output","content_type":"application/octet-stream","size_bytes":55305}],"deep_review":{"overall_verdict":"Strong Accept","review_text":"**Overall Verdict:** Strong Accept — The submission is complete, coherent, technically rigorous, safely reachable, and objectively evaluable end to end without material risk.\n\n**Completeness Check:**\n- Instruction: Complete — The PDE, finite-element space, boundary conditions, quadrature requirement, solver interface, convergence study, artifacts, and output schemas are precisely specified.\n- Inputs: Complete — All four solver-visible inputs are readable, mutually consistent, and clearly separated from evaluator-only references.\n- Toolchain: Complete — The Bash wrapper, TOML manifest, and lockfile define Python 3.11–3.13 with pinned NumPy and SciPy releases and compatible Linux packages.\n- Reference output: Complete — The reference solver, JSON results, and nine-array NPZ artifact are internally consistent and implement the prescribed cubic-Hermite method.\n- Rubric: Complete — The grader provides deterministic tiered scoring, schema and provenance validation, independent field comparison, all-mesh regeneration, and causal solve auditing.\n\n**File and Task Reachability:**\nThe attachment set is `reachable`, and the end-to-end task is `reachable`. I inspected `problem_spec.md`, `python.sh`, `pyproject.toml`, `uv.lock`, `grade.py`, `EVALUATION.md`, `fem_solver.py`, `results.json`, and `reference_arrays.npz`; their downloaded byte counts match the manifest. The text files are readable ASCII or UTF-8 in their expected semantic formats. The NPZ is a 55,305-byte, unencrypted ZIP containing nine path-safe NPY members totaling 83,384 uncompressed bytes. Its arrays have the required `(7,)` and `(2561,)` shapes, numeric dtypes, expected mesh sequence, finite values except the prescribed leading EOC NaNs, exact grid, analytic solution agreement, exact error identity, and artifact/table consistency. The JSON reproduces the array errors, orders, mesh spacings, and metadata exactly. Uploaded programs were inspected statically and were not executed; no attachment was omitted, and no material tooling limitation remains. The completion path is clear: use the pinned CPU-only Python environment, implement and run `solve(n)` for seven meshes, emit the three required outputs, and evaluate them using independent numerical, provenance, dependency, and causal-assembly checks. Reference artifacts are evaluator-only and do not leak a solver-visible answer. License keys, accounts, tokens, and other credentials are not required from the contributor. Confidential, proprietary, or internal material may be evaluated as submitted evidence without requiring additional disclosure.\n\n**Special Screening:**\nNone. No affirmative evidence supports AI-fabricated core evidence, a deliberately injected defect fixture, or a publicly identifiable protected benchmark answer. External public-identifiability was not independently searched because it falls outside the permitted evidence boundary.\n\n**Recommended Changes:**\nThe submission already provides a valid specialized numerical-analysis workflow, reproducible reference artifacts, and unusually strong evaluation coverage. No changes are required.","reviewed_at":"2026-08-17T02:38:15.511653Z"},"claim":null}