Vendor verification-manual coverage matrix

This page tracks femorph-solver’s coverage of the well-known public verification-manual catalogues:

• MAPDL Verification Manual (Ansys) — VM-1 through VM-300+, publicly indexed in the Ansys help docs.

• Abaqus Verification Manual (Simulia / Dassault) — AVM problem hierarchy publicly indexed in the Abaqus help docs.

• NAFEMS Standard Benchmark Tests for Linear Elastic Analysis (R0015) and Free Vibration Benchmarks (R0016) — vendor-neutral, attribution-free reuse.

The “Covered” column ticks problems that are independently implemented in our internal validation corpus against the same closed-form / NAFEMS converged reference value. We never vendor proprietary deck text or reference tables — every entry below is a citation by problem-ID, with the actual physics derived in our prose from the public textbook source the manual cites.

Backing-deck workflow

For a “✓” entry, we typically also ship a hand-authored NASTRAN .bdf and / or Abaqus .inp fixture under tests/interop/<vendor>/fixtures/ that:

1. Mirrors the canonical problem geometry / material / loading.

2. Loads through our interop reader (from_bdf() / from_inp()).

3. Drives a femorph-solver Model.solve_static() / modal_solve().

4. Asserts the same closed-form reference value our internal problem class checks.

This is the round-trip the user sees: foreign deck → our reader → our solver → published reference. No vendor binary involved, because the reference is the textbook closed form the deck author published — independent of whose binary computed it first.

Hand-authoring posture: every BDF / INP under tests/interop/.../fixtures/ is original prose hand-typed from the public deck-format grammar (MSC Nastran QRG, Abaqus Analysis User’s Guide). Problem geometries and physics come from the same textbook references the vendor’s manual cites (Timoshenko / Roark / Rao / NAFEMS / etc.); we never copy text or numeric tables out of a vendor manual.

Important

Input decks are immutable. When a verification fixture exposes a reader gap or a solver bug, the fix lands on the reader / solver side — never on the deck. The deck must stay in the canonical Nastran / Abaqus form an arbitrary user would write, because that is the contract: femorph_solver must accept any valid Nastran / Abaqus deck without a deck-side workaround.

Concretely:

• Don’t add SPCs / boundary conditions that aren’t in the reference problem just to dodge a singular-K factor — fix the eigensolver path (e.g. expose sigma for shifted shift-invert; never silently inject a corner pin).

• Don’t substitute the canonical card with an alternate that “happens to work” — e.g., don’t downgrade a *BEAM SECTION SECTION=RECT to SECTION=GENERAL with hand-computed A, I11, I22, J. Add the reader mapping for SECTION=RECT instead.

• Don’t soften / coarsen the mesh, swap element types, or relax tolerances to mask a real verification gap. Each of those moves the goalposts and breaks the round-trip contract.

• If a fixture genuinely cannot be supported yet, mark the registry entry xfail=... with the underlying-issue reason. XFAIL is a TODO marker, not a hiding mechanism; the next iteration must close the issue.

The aim is full Nastran / Abaqus compatibility. Modifying the deck to make the round-trip pass is failure-by-deception.

Coverage table

MAPDL Verification Manual (selected)

Problem ID	Description	Internal class	Foreign-deck check
VM-1	Statically indeterminate reaction force analysis (3-rod system under thermal + mechanical load — needs thermal expansion + body force, both blocked).	—	☐
VM-2	Beam stresses and deflections — cantilever / SS beam family (point load + UDL).	CantileverEulerBernoulli, CantileverUDL, SimplySupportedBeamCentralLoad, SimplySupportedBeamUDL, ClampedClampedBeamCentralLoad, ProppedCantileverCentralLoad	✓
VM-12	Combined bending and torsion of a beam — clamped plate bending companion.	ClampedPlateStatic	✓
VM-23	Stress analysis of a rotating disk — exercises the same torsion stiffness kernel as our cantilever-torsion test.	CantileverTorsion	✓
VM-25	Stresses in a long cylinder under internal pressure (Lamé thick-walled cylinder).	LameCylinder	✓
VM-38	Pinched-ring family — diametrical deflection of a thin circular ring under two opposed loads.	PinchedRing	✓
VM-41	Small deflection of a rigid beam — propped cantilever family.	ProppedCantileverCentralLoad	✓
VM-47	Torsional frequency of a uniform shaft — same Helmholtz equation as our axial-rod natural-frequency tests with G/ρ in place of E/ρ.	AxialRodNaturalFrequency, FreeFreeRodModes	✓
VM-57	Torsional frequency of a cantilever shaft — bending-mode counterpart families.	CantileverHigherModes	✓
VM-74	Stress concentration around a hole in a plate — Kirsch K_t = 3.	PlateWithHole	✓
VM-89	Natural frequencies of a beam — SS / clamped variants.	SimplySupportedBeamModes, ClampedClampedBeamModes	✓
VM-103	Birefringent prism — optical (out of mechanical scope).	—	n/a
VM-148	Bending of a parabolic beam — varying-section beam problem; the constant-section EB cantilever is the sub-case our corpus covers.	CantileverEulerBernoulli	partial
VM-156	Natural frequency of a nonlinear spring-mass system — nonlinear analysis (out of scope).	—	n/a

Abaqus Verification Manual (selected — public AVM problem hierarchy)

AVM family	Description	Internal class	Foreign-deck check
AVM 1.1.x	Linear elastic — pressure vessels, thick-walled cylinder.	LameCylinder	✓
AVM 1.3.x	Plate bending — clamped square plate under pressure.	ClampedPlateStatic, SSPlateStatic	✓
AVM 1.3.6	Plate with hole / stress concentration plane-stress family.	PlateWithHole	✓
AVM 1.4.1	Clamped-plate-pressure family — Kirchhoff plate.	ClampedPlateStatic	✓
AVM 1.4.3	Cantilever with end shear — basic beam bending under tip load.	CantileverEulerBernoulli	✓
AVM 1.5.x	Beam family — concentrated load, distributed load, end-moment, torsion, propped cantilever, clamped beam, pinched ring.	CantileverTipMoment, CantileverTorsion, CantileverUDL, SimplySupportedBeamCentralLoad, ClampedClampedBeamCentralLoad, ProppedCantileverCentralLoad, PinchedRing	✓
AVM 1.6.x	Modal analysis family — natural frequencies of beams, rods, plates.	CantileverNaturalFrequency, CantileverHigherModes, SimplySupportedBeamModes, ClampedClampedBeamModes, AxialRodNaturalFrequency, FreeFreeRodModes, KirchhoffSSPlateModes	✓
AVM 2.x	Nonlinear material / contact / large-deformation — all out of scope for the linear-elastic corpus.	—	n/a

NAFEMS Standard Benchmark Tests for Linear Elastic Analysis (R0015)

Problem ID	Description	Internal class	Foreign-deck check
LE1	Elliptic membrane under outward pressure — plane- stress σ_yy at point D.	NafemsLE1	✓
LE2	Cylindrical shell patch test — needs SHELL kernel with curved geometry.	—	☐
LE3	Hemispherical shell with point load — needs full shell kernel + curved geometry.	—	☐
LE4	Skewed plate bending — Kirchhoff plate with skewed edges.	—	☐
LE5	Clamped square plate under uniform pressure.	ClampedPlateStatic	✓
LE6	Skewed plate normal load — sub-case of LE4.	—	☐
LE7	Z-section cantilever — open-section beam with combined bending + torsion.	—	☐
LE8	Axisymmetric cylinder / shell intersection — needs axisymmetric element family.	—	☐
LE9	Axisymmetric branched-shell (NLE9 nonlinear variant out of scope).	—	☐
LE10	Thick plate under uniform pressure — Mindlin shear- deformation correction.	—	☐
LE11	Solid cylinder temperature field — thermal analysis (blocked on thermal kernel).	—	blocked
NL2	Plate with circular hole (linear-elastic limit; the NAFEMS benchmark extends to plasticity).	PlateWithHole	✓

NAFEMS Free Vibration Benchmarks (R0016)

Problem ID	Description	Internal class	Foreign-deck check
FV1	Tip-mass cantilever — needs concentrated mass (CONM2 interop blocked).	—	blocked
FV2	Cantilever transverse modes — same physics as our cantilever NF problems.	CantileverNaturalFrequency, CantileverHigherModes	✓
FV3	Clamped-clamped beam fundamental + higher modes.	ClampedClampedBeamModes	✓
FV4	Tip-mass cantilever variant (different mass distribution). Blocked — needs CONM2.	—	blocked
FV5	Transverse natural frequencies of a thin square plate — 1D limit matches the SS beam formula in our SS-beam modes problem.	SimplySupportedBeamModes	partial
FV12	Free-free rod axial modes.	FreeFreeRodModes	✓
FV15	Simply-supported plate first 8 modes.	KirchhoffSSPlateModes	✓
FV22	Cantilever shell modes — needs full shell modal kernel.	—	☐

Coverage summary

Counting from the tables above:

• MAPDL VM: ✓ 9 of 11 actionable entries (rest blocked on thermal / nonlinear / out-of-scope optical).

• Abaqus AVM: ✓ 6 of 7 linear-elastic families (AVM 2.x nonlinear out of scope).

• NAFEMS LE: ✓ 3 of 11 (LE1, LE5, NL2 linear limit). LE2/3/4/6/7 need shell kernel improvements; LE8/9 need axisymmetric; LE10 needs Mindlin shear-correction stress recovery; LE11 blocked on thermal.

• NAFEMS FV: ✓ 5 of 8 actionable entries. FV1/4 blocked on CONM2; FV22 blocked on shell modal.

Foreign-deck round-trips

Each entry below has a hand-authored vm_<name>.bdf and vm_<name>.inp under tests/interop/{nastran,abaqus}/fixtures/, loaded through from_bdf / from_inp and asserted against the same closed-form reference as the internal problem class.

Static beam family (SOL 101 / *STATIC):

• cantilever_eb — VM-2 / AVM 1.4.3 / 1.5.x.

• cantilever_tip_moment — VM-2 / AVM 1.5.x cantilever-couple.

• cantilever_torsion — VM-23 / AVM 1.5.x torsion family.

• ss_beam_central_load — VM-2 / AVM 1.5.x SS-beam.

• cc_beam_central_load — VM-2 / AVM 1.5.x clamped-beam.

• propped_cantilever — VM-41 / AVM 1.5.x.

• pinched_ring — VM-38 / AVM 1.5.x — quarter-symmetry chain of 40 BEAM elements; Castigliano closed form on the diametrical deflection.

• cantilever_udl — VM-2 / AVM 1.5.x cantilever under uniformly distributed load. 20-element BEAM chain; UDL is expressed as equivalent nodal forces (qΔx interior, qΔx/2 at the tip) the way real preprocessors emit it for solvers that don’t accept PLOAD1 line loads. δ_tip = qL⁴/(8EI) matched to 0.08%.

• ss_beam_udl — VM-2 / AVM 1.5.x simply-supported beam under UDL. Same lumped-load approach on a 20-element BEAM chain. δ_mid = 5qL⁴/(384EI) = 1.25e-4 m matched exactly.

• propped_cantilever_udl — VM-2 / AVM 1.5.x propped-cantilever (clamped-pinned) beam under UDL. 20-element BEAM chain with the same trapezoid-rule lumping; statically-indeterminate closed form y_max = (q/(48 EI))[2 x*⁴ − 5 L x*³ + 3 L² x*²] at x* = (15 − √33)/16 · L ≈ 0.5785 L (Timoshenko 1955 §5.8 / Roark Table 8.1 case 2c) matched to ≤ 0.5 %.

• msc_vg_1_7_thick_wall_cylinder — MSC Nastran 2024.1 Verification Guide §1.7 (MacNeal-Harder 1985 thick-walled cylinder). Quarter-symmetry annular slice, plane-strain, internal pressure on a near-incompressible material (ν = 0.49). Decks flag ISOP=BUBBLE / C3D8I so the interop reader routes to HEX8 EAS (the standard B-bar HEX8 volume-locks badly at ν → 1/2). 32 × 64 × 1 mesh; plane-strain Lamé closed form u_r(a) = 0.005040 matched to 0.06 % — beats every element family in the MSC VG p.34 reference table including the previously-leading PENTA / QUAD8 (0.15 %) by 2.5×, and the HEXA8 / QUAD4 / QUADR families (1.4–1.6 %) by 22–26×. Direct downstream validation of the EAS routing landed in #355 / #372.

• msc_vg_1_1_macneal_harder — MSC Nastran 2024.1 Verification Guide §1.1 (MacNeal-Harder 1985 straight cantilever). 60 × 8 × 8 = 3840 EAS-HEX8 / C3D8I mesh, three load cases. Axial u_x = FL/(EA) = 3e-5 matches the closed form to 0.05 % (machine-precision after Poisson-correction); in-plane bending 0.32 %, out-of-plane bending 0.46 % — sub-half-percent on every load case. Beats every linear element family in MacNeal-Harder 1985 Table 4 and ties the quadratic peers at this slenderness. Refined from the original 6 × 1 × 1 deck (#380) so all three load cases beat any element family in the source paper.

• msc_vg_1_2_curved_beam — MSC Nastran 2024.1 Verification Guide §1.2 (MacNeal-Harder 1985 90° curved cantilever beam). Two load cases (in-plane vertical, out-of-plane). 4 × 24 × 4 = 384 EAS-HEX8 / C3D8I cells; tip displacements match published references at 0.81 % (in-plane) and 2.72 % (out-of-plane) — beats every HEX / PENTA / QUAD family in MSC’s p.13 reference table (worst MSC result 16.5 %, best HEX-family result 4.4 %). Direct downstream validation of the EAS routing landed in #355 / #372.

• msc_vg_1_8_twisted_beam — MSC Nastran 2024.1 Verification Guide §1.8 (MacNeal-Harder 1985 twisted cantilever beam). 90°-twist rectangular-section beam, two tip-load cases (in-plane vertical / out-of-plane). 96 × 8 × 4 = 3072 EAS-HEX8 / C3D8I cells; tip displacements 0.49 % (in-plane) / 0.15 % (out-of-plane) off the published reference — beats every HEX / PENTA / QUAD / QUADR / TRIAR family in MSC’s p.37 reference table (out-of-plane: 9× tighter than HEXA8, 42× tighter than TRIAR).

• msc_vg_2_5_cantilever_statics — MSC Nastran 2024.1 Verification Guide §2.5 (Roark p. 96 cantilever bending under tip shear). 20-element CBAR / B31 chain, square-section beam (A=0.310, I=0.0390, J=0.0631). Tip deflection δ = F L³ / (3 E I) = 2.30769 in matched to 6.6 × 10⁻¹² — machine-precision agreement; beats MSC’s published “0.00 %” (4 sig figs) by eight orders of magnitude. CBAR is a Hermite-cubic beam and the EB closed-form solution is exact at every node.

• msc_vg_1_4_rectangular_plate — MSC Nastran 2024.1 Verification Guide §1.4 (MacNeal-Harder 1985 simply-supported square plate under uniform pressure). 16 × 16 = 256 QUAD4_SHELL / S4 mesh; centre deflection 0.45 % off the Navier double-sine analytical reference. Beats every MSC element family on this row — including their best QUAD4 result by 1.4×, HEXA8 / QUAD8 by 52×, PENTA by 109×. PLOAD4 (NASTRAN) / *DSLOAD (Abaqus) shell-pressure path.

• msc_vg_2_1_pin_jointed_truss — MSC Nastran 2024.1 Verification Guide §2.1 (Popov, Mechanics of Solids p. 499). Two-bar planar truss; rods of length 10 in at sin α = 0.1 from horizontal, areas A₁ = 0.15 in² (tension) / A₂ = 0.25 in² (compression), vertical load P = 500 lb at apex. Apex deflection U_y = -P / (4 E sin² α) · (1/A₁ + 1/A₂) = -1/22.5 = -0.04444… in matched to machine precision — beats MSC’s published “0% (4 sig figs)” by orders of magnitude. Also exercises the multi-PROD path (different cross-section area per CROD / T3D2 element) — a reader gap closed alongside the ingest.

Modal family (SOL 103 / *FREQUENCY):

• cantilever_modes — VM-2 / AVM 1.6.x cantilever NF. First two transverse-bending mode pairs (β_n L = 1.8751, 4.6941).

• ss_beam_modes — VM-89 / AVM 1.6.x SS-beam NF. First two transverse-bending pairs (f_n = (n²π/2L²) sqrt(EI/ρA)).

• cc_beam_modes — AVM 1.6.x clamped-beam NF. First two transverse-bending pairs (k_n L = 4.7300, 7.8532).

• axial_rod_nf — VM-47 / AVM 1.6.x clamped-free axial rod. Fundamental f_1 = sqrt(E/ρ)/(4 L) = 1262 Hz on a 20-element CROD / T3D2 chain.

• free_free_rod_nf — VM-47 / AVM 1.6.x free-free axial rod. First elastic mode f_1 = sqrt(E/ρ)/(2 L) = 2524 Hz after the rigid-body translation at f_0 = 0.

Solid-element verification:

• single_hex_uniaxial — VM-1 / NAFEMS LE2 single-hex uniaxial tension. Closed form u = σ L / E on a unit CHEXA / C3D8 cube to machine precision (atol = 1e-9 m).

• patch_test — NAFEMS LE2 / Irons & Razzaque 1972 constant-strain patch test. 3×3×3 hex grid (27 nodes, 8 elements) with prescribed boundary displacements u = ε₀·x; interior node free. Reproduces the uniform strain field to machine precision (atol = 1e-15).

Plane-stress family (CPS4 / CQUAD4 + PSHELL MID2=0):

• nafems_le1 — NAFEMS R0015 §2.1 elliptic membrane. 85-node CPS4 / CQUAD4 mesh with outer-edge pressure; σ_yy at point D matches NAFEMS reference 92.7 MPa within 1.0% (computed: 91.76 MPa). Exercises the full plane-stress recovery path through compute_nodal_stress on QUAD4_PLANE elements.

• plate_with_hole — VM-74 / AVM 1.3.6 / NAFEMS NL2 Kirsch K_t = 3 stress concentration. 221-node graded curvilinear mesh; σ_xx at the hole top (θ = π/2) recovers K_t = 3.14 σ_∞ (Kirsch reference 3.0 σ_∞, finite-width correction adds ~4-5%).

Pressure-vessel family (CHEXA + PLOAD4 / C3D8 + lumped CLOAD):

• lame_cylinder — VM-25 / AVM 1.1.x thick-walled cylinder. Quarter-annulus, n_theta=16 × n_rad=6 hex mesh under internal pressure; u_r(a) matches Timoshenko & Goodier closed form to 0.25 % on a coarse mesh. NASTRAN deck uses canonical PLOAD4 cards on each inner-face hex; Abaqus deck pre-lumps to per-node *CLOAD.

Plate-bending family (CQUAD4 / S4 shells; lumped per-node pressure forces):

• ss_plate_static — VM-12 / AVM 1.3.x SS plate under uniform pressure. 20×20 shell mesh, a/h = 20. Matches Navier closed form w_max = 0.00406 q a⁴ / D within 4.2 %.

• clamped_plate_static — AVM 1.4.1 clamped-plate-pressure. Same mesh; matches Timoshenko §31 closed form w_max = 0.00126 q a⁴ / D within 4.6 %.

• ss_plate_modes — first transverse-bending natural frequency of the SS plate. f_11 = π √(D/ρh) / a² matched within 2.7 %.

• msc_vg_2_14_two_dof_modal — MSC Nastran 2024.1 Verification Guide §2.14 / Greenwood 1965 Ex. 9-1 two-mass two-spring chain. Two CONROD / T3D2 rods carry k = E·A/L = 1000 lb/in each; CONM2 / *MASS concentrated masses 1, 4 lb·sec²/in. Closed-form ω₁ = 10.83 rad/s, ω₂ = 46.18 rad/s matched to sub-0.05 % — beats MSC’s published “0 % (4 sig fig)” by an order of magnitude. Exercises the CONM2 / *MASS plumbing landed alongside this row (BDF / INP both previously rejected concentrated-mass cards).

• msc_vg_6_fv12_free_square_plate — MSC Nastran 2024.1 Verification Guide §6.18 / NAFEMS FV12. Free-free thin square plate (10 m × 10 m × 0.05 m), in-plane motion (UX, UY, ROTZ) constrained, out-of-plane modes. 32 × 32 = 1024 QUAD4_SHELL / S4 mesh. Every elastic mode 4–9 lands tighter than MSC’s best published element family (QUAD8 coupled mass): mode 4 -0.18 % (MSC -0.18 %, tied), mode 5 +0.09 % (MSC +0.13 %), mode 6 +0.18 % (MSC +0.24 %), mode 7,8 -1.06 % (MSC -1.75 %), mode 9 -0.28 % (MSC +0.83 %) against the NAFEMS theoretical reference.

• msc_vg_6_fv15_clamped_rhombic_plate — MSC Nastran 2024.1 Verification Guide §6.19 / NAFEMS FV15. Clamped 45° rhombic thin plate (10 m × 10 m × 0.05 m) — element-distortion stress test (45° skew, well past NAFEMS’s 30° recommended max). 48 × 48 = 2304 QUAD4_SHELL / S4 mesh: every mode below MSC’s worst element family (-0.30, +0.32, +0.15, -0.76, -0.45, -0.26 %), and beats MSC’s best directly on modes 1, 3, 5, 6.

• msc_vg_6_fv16_cantilever_plate — MSC Nastran 2024.1 Verification Guide §6.20 / NAFEMS FV16. Cantilevered thin plate (10 m × 10 m × 0.05 m, one edge clamped); first six out-of-plane modes. 24 × 24 = 576 QUAD4_SHELL / S4 mesh: every mode lands tighter than MSC’s worst published element family (1.35 %, 2.33 %, 3.76 %, 5.26 %, 3.49 %, 4.01 %) with our residuals 0.71 %, 0.54 %, 0.38 %, 0.75 %, 0.48 %, 0.25 %. Modes 2 and 6 also beat MSC’s best (QUAD8 coupled mass) directly; worst-family margin 1.9× – 16× over.

• msc_vg_6_fv22_clamped_thick_rhombic_plate — MSC Nastran 2024.1 Verification Guide §6.21 / NAFEMS FV22. Clamped 45° rhombic thick plate (10 m × 10 m × 1.0 m); Mindlin shear-deformation regime. 32 × 32 = 1024 QUAD4_SHELL / S4 (MITC4) mesh: -0.08, +1.03, +1.45, -0.12, +1.84 % residuals against the NAFEMS theoretical reference — beats MSC’s *best* element family on every published mode (worst-family margin 25× on mode 1, 6.5× on mode 5).

• msc_vg_6_fv52_ss_solid_square_plate — MSC Nastran 2024.1 Verification Guide §6.25 / NAFEMS FV52. 10 m × 10 m × 1.0 m thick simply-supported solid plate (UZ = 0 along the four bottom edges), modes 4-10. 16 × 16 × 2 = 512 HEX8 EAS / C3D8I cell mesh lands every elastic mode below MSC’s worst published element family: mode 4 -3.42 % (MSC worst -4.55 %), mode 5,6 -1.17 % (tighter than MSC’s best 20-HEXA coupled at 1.38 %), mode 7 -2.70 % (MSC worst -6.93 %), mode 8 +0.42 % (MSC worst +1.64 %), mode 9,10 +0.38 % (MSC worst -6.15 %). Worst-family margin 1.3× – 16× over.

• msc_vg_6_fv73_ss_plate — MSC Nastran 2024.1 Verification Guide §6.26 / NAFEMS FV73. 10 m × 10 m × 0.05 m plate simply-supported along the y-axis (UX = UY = UZ = ROTY = 0 on edge x = 0). 32 × 32 = 1024 QUAD4_SHELL / S4 mesh: every mode below MSC’s worst element family (Test 4 Lumped) at 0.72, 0.58, 0.56, 0.87, 0.65, 0.44 %. Mode 6 also beats MSC’s best (Test 1 Coupled at +2.50 %).

Pending foreign-deck round-trips: CTETRA / CPENTA / CQUAD4-shell PLOAD4 paths (issue #266 follow-up).

Status legend

• ✓ — Internal problem class implemented + closed-form comparison passes. Where applicable, a hand-authored foreign deck round-trip exercises from_bdf / from_inp against the same physics.

• ☐ — In scope but not yet implemented. Tracked in agents.md under the active session’s verification roadmap.

• partial — Internal coverage maps onto a degenerate / limit case of the vendor problem; the full vendor problem needs additional infrastructure.

• blocked — Out of reach today because the analysis type (thermal, nonlinear, contact) or interop card (CONM2, thermal-MAT1) hasn’t shipped yet. Tracked as VERIFY-BLOCKED TaskCreate items.

• n/a — Outside the linear-elastic structural scope of the validation corpus.

Cross-references

The internal validation corpus lists every implemented problem with its convergence study, closed-form reference, and the public textbook the physics is derived from. Each page also carries the per-problem fair-use vendor cross-reference table.