# Linear Static 3D Euler Beam Research Brief

## Metadata

- feature_id: `linear-static-3d-euler-beam`
- source_requirement: `docs/linear-static-3d-euler-beam/requirements.md`
- approved_design: `docs/superpowers/specs/2026-08-08-linear-static-3d-euler-beam-design.md`
- source_formulation: `docs/linear-static-3d-euler-beam/formulation.md`
- status: `ready-for-formulation`
- owner_agent: `research-agent`
- date: `2026-08-09`
- evidence_route: configured FEM wiki at `C:\git\MultiPhysicsVault`; `wiki/hot.md` -> `wiki/index.md` -> selected wiki pages and their recorded raw-source chunks
- source_policy: verified external facts use Tier 1 or Tier 2 sources; approved FESA decisions and mathematical consequences are labeled `Project contract` or `Inference`

This brief supplies evidence for formulation alignment and later verification planning. It does not approve the existing formulation, define a C++ API, introduce a tolerance, or claim full Abaqus compatibility.

## Research Questions

1. What distinguishes Abaqus B31 from B33, and what supports treating only B33 as the Euler–Bernoulli member in FESA V0?
2. What evidence supports cubic Hermite bending interpolation, six free-element rigid modes, rank 6, two-point Gauss exactness, local/global transformation, and end-force sign checks for a two-node spatial Euler beam?
3. What do Abaqus general beam section data `A, I11, I12, I22, J`, the first section axis `n1`, beam resultants `SF*`/`SM*`, and axial `S11` mean?
4. Which axial, torsional, and two-plane bending cantilever cases provide analytical checks, and which quantities or behaviors do they not verify?
5. What technical evidence and limitations apply to SPD factorization, PARDISO selection, deterministic sparse assembly, and row-based reference comparison?

## Source Inventory

| id | document | provider_or_author | wiki page identity | original source/manual identity | reliability tier | applicable evidence |
| --- | --- | --- | --- | --- | --- | --- |
| S1 | Abaqus beam element and section documentation | Dassault Systèmes / SIMULIA | [[Abaqus Structural Element Families]]; [[Abaqus Beam and Shell Section Definitions]] | *Abaqus 2016 Analysis User's Guide, Volume IV: Elements*, sections 29.3.3, 29.3.4, 29.3.7, 29.3.8; vault chunks `AbaqusAnalysisUserGuide4_030.md`, `_031.md`, `_033.md`, `_035.md` | Tier 1 | B31/B33 family and interpolation classification, slender-beam limit, section axes/properties, beam output components |
| S2 | Beam and frame finite elements | Daryl L. Logan; Thomson | [[Beam and Frame Finite Elements]] | *A First Course in the Finite Element Method*, Fourth Edition, Chapter 4 and Gaussian quadrature material; chunks `AFirstCourseInTheFiniteElementMethod_018.md`, `_020.md`, `_049.md` | Tier 2 | Euler–Bernoulli assumptions, cubic Hermite interpolation, end-force signs, equivalent loads, cantilever checks, Gauss exactness |
| S3 | Static equation solution | Klaus-Jürgen Bathe | [[Static Equilibrium Equation Solvers]] | *Finite Element Procedures*, Second Edition (2014), Chapter 8; chunks `FiniteElementProcedures_072.md`, `_074.md`, `_076.md`, `_077.md` | Tier 2 | symmetry/positive-definiteness conditions, LDLT/Cholesky factorization, positive-pivot failure, conditioning limits |
| S4 | Isoparametric mapping and quadrature synthesis | FEM wiki, with Bathe, Logan, Abaqus, and local source provenance | [[Isoparametric Finite Elements]] | *Finite Element Procedures*; Logan textbook; *Abaqus Theory Manual*; *Abaqus Analysis User's Guide, Volume IV* | Tier 2 synthesis; Tier 1 where Abaqus-specific | natural coordinates, Jacobian mapping, quadrature choice, under-integration risk |
| S5 | FEM program verification workflow | FEM wiki, with textbook and official-manual provenance | [[Finite Element Program Implementation]]; [[Finite Element Modeling and Convergence Checks]] | *Finite Element Procedures*; Logan textbook; Abaqus user guides; production solver manuals | Tier 2 synthesis; Tier 1 for cited official-manual behavior | element-local/global separation, sparse assembly context, constraint/solve/recovery workflow, benchmark and equilibrium checks |
| P1 | Approved FESA feature contract | FESA project | N/A | approved requirements and design named in Metadata | Project contract, not external evidence | exact V0 scope, fixed tolerance policy, read-only reference inventory, required output and orchestration |
| P2 | Existing candidate formulation | FESA project | cites [[Beam and Frame Finite Elements]], [[Isoparametric Finite Elements]], and S5 pages | `docs/linear-static-3d-euler-beam/formulation.md` | Candidate derivation, not an approved source | equations and sign choices that downstream Formulation and Numerical Review agents must check |

The wiki source records identify S1 as high-confidence manual provenance, S2 as a high-confidence textbook source, and S3 as a current textbook source. P1 and P2 are intentionally not assigned an external reliability tier.

## Extracted Facts

### B31, B33, and the supported Euler–Bernoulli meaning

- **F-01 — Verified, Tier 1 (S1):** Abaqus lists B33 as a two-node cubic three-dimensional beam and includes B33/B33H among its Euler–Bernoulli beam elements. These elements suppress transverse shear deformation and retain cross-sections normal to the beam axis in the no-warping idealization. [[Abaqus Structural Element Families]]
- **F-02 — Verified, Tier 1 (S1):** Abaqus lists B31 as a two-node linear three-dimensional Timoshenko beam. B31 allows transverse shear deformation, so interpreting B31 as the same formulation as a classical shear-free B33 changes the element meaning. [[Abaqus Structural Element Families]]
- **F-03 — Verified, Tier 1 (S1):** The manual gives a modeling guideline, not a universal acceptance threshold: for a homogeneous member, a typical cross-section dimension below roughly `1/15` of the typical axial distance makes transverse-shear flexibility negligible. [[Abaqus Structural Element Families]]
- **F-04 — Project-contract conclusion (P1), supported by F-01/F-02:** Mapping only `TYPE=B33` to the FESA Euler–Bernoulli element and rejecting B31 preserves the documented family distinction. This establishes a supported subset; it does not establish numerical or file-format equivalence with every Abaqus B33 capability.
- **F-05 — Applicability distinction, Tier 1 plus P1:** Abaqus documents its cubic beam for a wider small-strain/large-rotation setting, whereas FESA V0 deliberately restricts the model to small displacement and small rotation. The FESA restriction is narrower than the source element family. [[Abaqus Structural Element Families]]

### Interpolation, modes, integration, transformation, and signs

- **F-06 — Verified, Tier 2 (S2):** A two-node Euler–Bernoulli bending field has transverse displacement and slope/rotation at each node. A complete cubic field supplies the four interpolation conditions and is `C1` across connected beam elements; the functions are Hermite cubic interpolation functions. [[Beam and Frame Finite Elements]]
- **F-07 — Verified, Tier 2 (S2):** Gaussian quadrature with `n` points integrates a polynomial through degree `2n-1` exactly; two points therefore integrate a cubic polynomial exactly. [[Isoparametric Finite Elements]]
- **F-08 — Inference from F-07 and P2:** For the straight, constant-property FESA candidate, the bending rows of `B` are linear in the natural coordinate, so each entry of `B^T D B J` is at most quadratic. Two-point Gauss integration should reproduce the closed-form stiffness. This is a testable consequence, not final approval of the current `B` matrix.
- **F-09 — Verified principle, Tier 2 (S2/S4):** Spatial frame members transform nodal vectors between local member axes and the global system before assembly. An orthonormal, right-handed rotation preserves vector work; FESA energy-invariance and orthogonality checks are downstream consequences. [[Beam and Frame Finite Elements]]
- **F-10 — Inference from P2 and the 12-DOF model:** A free two-node spatial beam should have three translational and three rotational rigid-body modes. With six independent positive deformation modes, its 12-by-12 stiffness should have rank 6. The queried wiki does not state the exact FESA rank claim, so all six null vectors, six positive modes, and scale-aware rank classification require independent Numerical Review and executable tests.
- **F-11 — Verified sign risk, Tier 2 (S2):** Textbook beam nodal force and classical section-cut conventions can have opposite signs at an element end; local end forces must be defined as forces exerted on the element or on the adjoining cut, not inferred only from a component name. [[Beam and Frame Finite Elements]]
- **F-12 — Project-contract inference (P1/P2):** The approved FESA endpoint normalization and `theta_y=-dw/dx` determine the proposed end-action signs. S1 defines Abaqus output component meanings but does not prove the exact FESA-to-Abaqus endpoint section-cut sign transform. That transform remains an explicit review and reference-test item.

### Abaqus general beam section and output meanings

- **F-13 — Verified, Tier 1 (S1):** For `*BEAM GENERAL SECTION, SECTION=GENERAL`, the manual orders the direct geometric data as `A, I11, I12, I22, J` before optional warping-related terms. `A` is area, `I11` and `I22` are moments of inertia for bending about section axes 1 and 2, `I12` is the cross-bending inertia, and `J` is the Saint-Venant torsional constant. [[Abaqus Beam and Shell Section Definitions]]
- **F-14 — Verified, Tier 1 (S1):** Abaqus defines a local right-handed `(t,n1,n2)` beam section frame: `t` is positive from the first element node to the second, `n1` is the first beam section axis, and `n2` is the second section axis/beam normal. For a spatial beam, an approximate `n1` can be supplied with the section or an off-axis node. [[Abaqus Beam and Shell Section Definitions]]
- **F-15 — Mathematical inference from F-14 and Project contract P1:** In a right-handed frame, `n2=t x n1`; therefore the approved mapping `t -> local x`, `n1 -> local y`, and `t x n1 -> local z` is consistent with the manual axis order. `Iy=I11`, `Iz=I22`, and rejection of `I12!=0` are the approved FESA subset and must not be generalized to all Abaqus general sections.
- **F-16 — Verified, Tier 1 (S1):** Abaqus beam output defines `SF1` as axial force, `SM1` as bending moment about local section axis 1, `SM2` as bending moment about local section axis 2, and `SM3` as twisting moment about the beam axis. `SF2` and `SF3` are transverse shear components and are unavailable for B33/B33H. [[Abaqus Structural Element Families]]
- **F-17 — Verified, Tier 1 (S1):** `S11` is axial beam stress. Shear components documented for other beam/section cases do not establish that FESA V0 can recover them. [[Abaqus Beam and Shell Section Definitions]]
- **F-18 — Project-contract mapping (P1), supported by F-14/F-16:** With section axis 1 mapped to FESA local `y` and axis 2 to local `z`, the approved legacy comparison uses `SF1 -> N`, `SM1 -> My`, `SM2 -> Mz`, and `SM3 -> T`. The transverse `SF2/SF3` rows are outside the approved CSV comparison.

### Static solution, sparse assembly, and comparison evidence

- **F-19 — Verified, Tier 2 (S3):** Properly supported linear elastic FE systems commonly produce symmetric positive-definite coefficient matrices. Cholesky requires positive definiteness, and a nonpositive pivot indicates an invalid assumption, insufficient support, indefiniteness, or severe numerical difficulty rather than a result to accept silently. [[Static Equilibrium Equation Solvers]]
- **F-20 — Verified, Tier 2 (S3):** Matrix factorization and load-vector reduction/back substitution are distinct stages; retaining a factorization permits additional right-hand sides without repeating decomposition. [[Static Equilibrium Equation Solvers]]
- **F-21 — Project-contract inference (P1):** Using an SPD PARDISO mode for constrained `Kff` is consistent with F-19 only after positive section/material data and sufficient constraints remove all rigid modes. The FEM vault contains no Intel oneMKL PARDISO manual page, phase contract, or CSR indexing contract, so those product-specific details are not verified here.
- **F-22 — Verified workflow, Tier 2 synthesis (S5):** Reliable FEM software separates element-local calculations, global DOF mapping and assembly, constraint application, equation solution, and result recovery, then checks new elements with analytical or benchmark problems. [[Finite Element Program Implementation]]
- **F-23 — Project-contract inference (P1):** Stable element ordering, deterministic triplet sorting, duplicate reduction, and canonical 0-based CSR finalization are FESA reproducibility policies. The queried wiki supports sparse assembly as a stage but does not prescribe this exact deterministic algorithm.
- **F-24 — Verified verification principle, Tier 2 synthesis (S5):** Displacement, reaction, equilibrium, compatibility, convergence, coordinate-system meaning, and result location must be checked before trusting output tables. [[Finite Element Modeling and Convergence Checks]]
- **F-25 — Project contract (P1):** Exact row identities, the component-scale mixed tolerance, all-row reporting, and fail-fast handling of missing/extra/nonfinite rows come from approved FESA requirements. This brief neither alters nor independently approves those numerical values.
## Candidate Benchmarks

The equations below reproduce the already approved analytical candidates in P1/P2. They are not newly generated reference values. Load and response signs assume the P2 local-axis and rotation convention and must be independently checked before formulation approval.

| benchmark_id | evidence classification | configuration | target quantities | can verify | cannot verify |
| --- | --- | --- | --- | --- | --- |
| `CB-AXIAL-TIP` | Tier 2 beam/bar principle plus P2 equation | one straight cantilever, tip force `Px` along local `+x` | `u(L)=Px L/(EA)`, fixed-end reaction, constant `N`, axial `S11` at specified section points | axial stiffness, DOF/component direction, reaction equilibrium, centroidal axial stress | bending, torsion, B33 interpolation details, reference row normalization |
| `CB-TORSION-TIP` | Tier 2 member principle plus P2 equation | one cantilever, tip torque `Mx` about local `+x` | `theta_x(L)=Mx L/(GJ)`, reaction torque, constant `T` | Saint-Venant torsional stiffness, rotation sign, reaction and section torque | warping, bimoment, torsional shear-stress distribution, open-section fidelity |
| `CB-BEND-LOCAL-Y` | Tier 2 Euler beam theory plus P2 equation | one cantilever, tip force `Py` along local `+y` | `v(L)=Py L^3/(3 E Iz)`, `theta_z(L)=Py L^2/(2 E Iz)`, reaction force/moment, linear `Mz` | Hermite bending plane, `Iz=I22`, `theta_z` sign, endpoint equilibrium | shear deformation, deep-beam behavior, transverse stress recovery |
| `CB-BEND-LOCAL-Z` | Tier 2 Euler beam theory plus P2 sign convention | one cantilever, tip force `Pz` along local `+z` | `w(L)=Pz L^3/(3 E Iy)`, `theta_y(L)=-Pz L^2/(2 E Iy)`, reaction force/moment, linear `My` | second bending plane, `Iy=I11`, critical `theta_y=-dw/dx` sign | shear deformation, `I12` coupling, transverse stress recovery |
| `CB-RIGID-RANK` | Inference F-10 | unconstrained, otherwise valid free element | six explicit rigid vectors, `K r_i`, numerical rank, positive energy of six deformation modes | missing/extra constraints, sign errors that create spurious strain, rank loss from integration | global constraint sufficiency, PARDISO behavior, deep-beam accuracy |
| `CB-GAUSS-CLOSED` | Tier 2 Gauss theorem plus inference F-08 | constant `E,G,A,Iy,Iz,J`, straight element | two-point-Gauss stiffness versus independently evaluated closed form | quadrature/Jacobian/`B` consistency and stiffness symmetry | variable property/load integration, curved or tapered members |
| `CB-ROTATED-SPACE` | Tier 2 frame transformation plus inference F-09 | rigidly rotate each analytical cantilever to a non-axis-aligned 3D orientation | transformed displacement/reaction/resultant, `R R^T`, determinant, strain energy | local-axis construction, local/global mapping, energy invariance | instance transforms, curved axes, orientation fallback outside P1 |
| `CB-B31-NEGATIVE` | Tier 1 S1 plus P1 | otherwise valid input with `TYPE=B31` | `unsupported-element-formulation`; no Euler element | semantic preservation of Abaqus family meaning | numerical B31/Timoshenko behavior |
| `CB-LEGACY-B33-ROWS` | Project contract P1 with Tier 1 output semantics | exact read-only `reference/cantilever beam/` B33 bundle | displacement, reaction, node-station section-resultant rows | supported pipeline quantities, identities, component meanings, approved tolerance application | beam stress comparison, `SF2/SF3`, broad Abaqus compatibility, correctness outside this model |

## Verification Relevance

- **code_verification:** `CB-RIGID-RANK`, `CB-GAUSS-CLOSED`, Hermite value/slope conditions, stiffness symmetry, and `CB-ROTATED-SPACE` isolate interpolation, differentiation, integration, transform, and sign errors before reference comparison. Evidence basis: F-06 through F-12.
- **solution_verification:** The four analytical cantilevers compare exact member quantities against numerical displacement, rotation, reaction, section resultant, residual, and axial stress. Separate load cases prevent one sign or stiffness error from being hidden by a mixed response. Evidence basis: S2 and P2.
- **solver_verification:** `Kff` symmetry/positive-definiteness, factorization success, repeated-RHS substitution, normalized residual, and reaction recovery check the linear system path. A PARDISO-specific claim still requires the official oneMKL contract identified in Open Issues. Evidence basis: F-19 through F-21.
- **assembly_reproducibility:** Repeating assembly with changed thread counts should produce the same canonical CSR structure and values required by P1. This is a project-policy test, not an externally sourced theorem. Evidence basis: F-22/F-23.
- **reference_comparison:** S1 establishes Abaqus `U/RF/RM/SF/SM` component meaning; P1 establishes exact files, row identities, component mapping, and tolerance. Artifact/schema mismatches must fail before numeric comparison. `S11` comparison is explicitly N/A.
- **physics_sanity:** Global force/moment balance, reaction sign, displacement direction, two bending-plane symmetry, endpoint section-force consistency, and free-DOF residual remain separate from numerical similarity to the CSV. Evidence basis: F-24 and P1.
- **validation:** No experimental or physical-validation dataset was identified. These candidates support code verification, analytical solution verification, and source-solver comparison only; they do not validate Euler–Bernoulli theory for a real deep, warped, curved, nonlinear, or anisotropic member.

## Applicability Limits

- **linear_or_nonlinear:** linear static only; no geometric or material nonlinearity.
- **deformation:** FESA small displacement and small rotation. Abaqus B33's broader documented rotation capability is not inherited.
- **element_type:** straight, prismatic, two-node FESA Euler–Bernoulli element mapped only from `TYPE=B33`; no B31/Timoshenko reinterpretation.
- **slenderness:** Euler–Bernoulli neglects transverse shear. The S1 `1/15` statement is a modeling guideline and is not converted here into a FESA warning or rejection threshold.
- **section:** `A,I11,I12,I22,J` general beam section data with approved `I12=0`; no centroid/shear-center offset, warping constants, taper, release, or curved member.
- **material_model:** homogeneous isotropic linear elasticity with positive approved properties; no plasticity, anisotropy, temperature dependence, or nonlinear section integration.
- **loads:** nodal concentrated loads in the CLI. The constant local line-load vector is only a formulation verification kernel; it does not imply `*DLOAD` support.
- **integration:** two-point exactness applies to the straight constant-property candidate whose integrand degree is covered by F-08. It is not a general rule for variable properties, curved geometry, discontinuities, or higher-order loads.
- **coordinate_system:** right-handed local frame based on tangent and supplied `n1`; no silent fallback for zero or tangent-parallel guide vectors and no instance transform.
- **solver:** SPD factorization applies only to valid, sufficiently constrained `Kff`. Unconstrained element matrices are positive semidefinite and singular; indefinite, nonsymmetric, ill-conditioned, or multiplier systems are outside this contract.
- **results:** displacement/reaction, equilibrium end action, section resultants, generalized results, and axial `S11` only. Transverse/torsional shear stress and Abaqus stress comparison are outside scope.
- **units:** user-consistent units generally; the approved legacy comparison alone carries the external SI contract from P1.
- **reference_scope:** the existing B33 cantilever bundle is read-only and verifies only documented quantities and model coverage. It cannot establish broad B33 compatibility.

## Open Issues

1. **PARDISO manual gap — non-blocking for formulation:** The configured FEM wiki has no Intel oneMKL PARDISO source page. Exact SPD matrix type, phase sequence, zero-based CSR option, reordering, repeated-RHS lifetime, and error-code handling must be checked against an official Intel manual before implementation planning calls the adapter contract source-backed.
2. **Deterministic COO-to-CSR gap — project policy:** S5 supports the assembly stage, but no queried source prescribes FESA's stable element order, triplet sort key, duplicate summation order, or thread-count-independent reduction. The implementation plan must specify and test them without presenting them as textbook facts.
3. **Endpoint sign-normalization gap:** S1 defines `SF*`/`SM*` meanings and S2 warns that nodal end-action and section-cut signs differ. The exact FESA endpoint-to-node-station transform must be re-derived in Formulation Alignment, checked independently in Numerical Review, and exercised against unmodified reference rows.
4. **Exact Abaqus B33 equivalence is not established:** The manual describes a two-node cubic Euler–Bernoulli family and notes additional internal axial variables for cubic beams. This brief supports the approved physical subset, not internal-algorithm identity or bitwise equivalence.
5. **Slenderness policy remains intentionally unset:** The official `1/15` guidance documents applicability but P1 contains no automatic slenderness diagnostic. Adding one requires a requirements amendment.
6. **Legacy row-location semantics:** The approved CSV provides node-station force rows rather than complete element-end identity. Step 5 must retain the interior-endpoint consistency check and deterministic representative selection; averaging cannot hide a sign or continuity mismatch.

None of these issues changes the approved V0 scope or blocks mathematical formulation alignment. Issues 1 and 2 are mandatory implementation-planning handoffs; Issues 3 and 4 are mandatory for formulation and numerical review.
## Downstream Handoff

### Formulation Agent

- Use F-01 through F-18 to align the formulation identity to B33's shear-free cubic member meaning while retaining the narrower small-rotation FESA scope.
- Re-derive both bending-plane signs, especially `theta_y=-dw/dx`, and make equilibrium end action versus section-cut resultant conventions explicit at both endpoints.
- Treat six rigid modes/rank 6, two-point exactness, and `n2=t x n1` as reviewable consequences, not as facts made true merely by this brief.
- Do not add shear deformation, `I12` coupling, offsets, warping, or new tolerances.

### Numerical Review Agent

- Independently check all six rigid vectors, numerical rank scaling, positive deformation energy, stiffness symmetry, and Gauss/closed-form agreement.
- Check transformation orthogonality, determinant, virtual-work/energy invariance, and both endpoint signs using a rotated 3D member.
- Confirm `SM1/SM2` mapping after `n1 -> y`, `n2 -> z`, and audit equilibrium end action versus section resultant.
- Preserve S1 slenderness guidance as an applicability limit unless requirements are amended.

### I/O Definition Agent

- Use S1 for exact meanings of `A,I11,I12,I22,J`, `n1`, `S11`, `SF*`, and `SM*`, while documenting only the P1-approved keyword subset.
- Keep optional Abaqus warping/offset data and B31 outside semantic mapping rather than silently discarding or approximating them.
- Keep result location and coordinate metadata explicit enough to distinguish endpoint actions, section resultants, and Gauss/section-point `S11` rows.

### Reference Model Agent

- Cover axial, torsional, local-y bending, local-z bending, and rotated-space checks with target quantities and exclusions from Candidate Benchmarks.
- Preserve exact legacy B33 paths and do not generate, repair, rename, or rewrite reference artifacts.
- Make stress comparison N/A, `SF2/SF3` noncoverage, and interior endpoint consistency visible in the coverage matrix.

### Implementation Planning Agent

- Translate every candidate into a RED/GREEN/VERIFY test without changing P1 tolerances.
- Resolve Open Issues 1 and 2 with an official oneMKL contract and explicit deterministic reduction algorithm before planning PARDISO and sparse-assembly implementation.
- Keep factorization, load assembly, effective-RHS formation, and substitution observable as separate events.

### Coordinator Agent

- Carry the PARDISO source gap and endpoint-sign review forward. Reopen requirements only if a new slenderness diagnostic, broader Abaqus compatibility claim, or changed result/tolerance contract is proposed.
