Validation dossier

FEXLA runs its own finite-element engine, written by us in TypeScript and run in your browser. The model is built, meshed and solved on your computer. This dossier records how each part of that engine, and the WRC 537 engine behind the local-stress calculator, has been checked: against closed-form solutions, against ASME PTB-3 worked examples, and against three independent open-source finite-element programs, CalculiX, code_aster and MYSTRAN.

Mesh sizing. Next to each junction the first element is the smaller of the wall thickness and a tenth of λ = √(R·t), inside a refined zone of 2.5λ, growing to the smaller of λ and a tenth of R in the far field.

The equation solver was checked by exporting six assembled systems, of 25,717 to 138,331 equations, and solving them again in two independent sparse direct solvers. The answers agree at the round-off floor for the condition numbers involved.

That check ran on the earlier up-looking factorisation. The multifrontal factorisation now in use is checked against dense elimination in the test suite (record T15); the six-system comparison has not been repeated on it (chapter 15).

How to read the comparisons. In most chapters every program solves exactly FEXLA's model: the same mesh, the same supports and FEXLA's own assembled nodal loads, so a difference comes from the element and the solver, not from the modelling. Where the programs used independent meshes, the chapter says so. "Median element difference" is the median, over the elements carrying at least a quarter of the peak stress, of the difference in surface von Mises stress at the element centre.

Each table row names its record. The record list at the end gives the file each record is kept in, in the FEA repository, in this site's repository, or as a study record. Programs used: CalculiX 2.22, code_aster 17.4.0, MYSTRAN 19.0.0, Gmsh 4.15.2, SciPy 1.18.1 (SuperLU).

What these comparisons are, and are not. They are scripted runs made by the development team and recorded in the reports cited. The verification register also asks for a hand check and a cross-check signed by a named engineer for each feature; those sign-offs are still open (record P02, chapter 15).

Before any comparison with another program, the element and the assembled model must reproduce answers known in closed form: the patch tests, the membrane state of a cylinder, the shell benchmarks of Belytschko and of MacNeal and Harder, thermal stress and an offset section.

The rectangular box module (VIII-1 Appendix 13) has closed forms of its own. The shell model reproduces them at the same dimensions:

The brick element has its own closed-form checks (patch test to 1e-8, a cantilever against Timoshenko, a thick ring against Lamé); they are in chapter 10.

Local bending at a junction is what a shell model is for. Three cases have answers that do not need another program: a cylinder clamped at its end, a cylinder with a step in thickness, and the knuckle of a formed head checked against an axisymmetric solid.

The knuckle reads 5 to 7 % below the solid. The report traces it to the shell idealisation of the head's mid-surface; it is listed as a stated tolerance, to be rechecked on a torispherical head (chapter 15).

The first comparison wrote FEXLA's mesh and nodal loads to CalculiX in two forms: S4 (expanded to C3D8I bricks) and S8R (expanded to C3D20R, mid-side nodes on the flat facets). Displacements agree within about 1 %; FEXLA's peak surface stress lies between the two CalculiX elements in every case.

The I-beam clip's S8R peak is at the plate corner on the shell, where the linear-elastic stress is singular and every element type reports a different finite number. It needs a weld-toe or structural-stress reading, not refinement.

The solver check of 2026-10-18 repeated this for every shell capability, with code_aster's Q4G shell (the same MITC4 family, independently written) as well as CalculiX S4. code_aster applies the face pressure itself, so FEXLA's consistent pressure loads are checked too.

Several attachments on one shell. With two nozzles on one cylinder the three formulations (FEXLA, CalculiX S4, S8R) agree on every interaction ratio to within 0.7 % (0.3 % on the finer mesh). On a head and on a flat closure the comparison used independent meshes: CalculiX on Gmsh meshes of second-order triangles, sharing nothing with FEXLA but the geometry, material and loads.

A published sample from commercial nozzle design software (a hillside nozzle with a pad, moment and force range) was re-run in FEXLA and in CalculiX. The shell side at the junction agrees with the published value within 0.2 % at the default mesh, though it rises 13 % with refinement, so that agreement is partly coincidence. On the nozzle side FEXLA is mesh-converged at 54.5 ksi and CalculiX S4 agrees within 1.7 %, but the published value is 105 ksi, about twice. The sample does not show the modelling choices behind that number; the difference is open (chapter 15).

MYSTRAN 19.0.0 solves the same models in bulk-data form: CQUAD4 shells, CHEXA bricks and RBE2 rigid elements. Its quad has no drilling stiffness, so it is the most different of the three shell elements from FEXLA's.

On four of the five shell models MYSTRAN agrees within 0.3 to 1.9 % (median). On the nozzle on a cylinder with combined end loads its displacement is 3.7 % larger and its element differences wider; code_aster is 2.4 % more flexible there too, while CalculiX is within 0.85 %. Two independent elements agree with each other on that one case and not with FEXLA and CalculiX. The likely place is where the nozzle wall meets the shell at right angles, where each element stiffens the in-plane rotation its own way. It is open item Q11 (chapter 15).

MYSTRAN's brick has no incompatible modes, so it is 1.9 % stiffer in the cantilever; on the thick ring it agrees within 0.06 %.

ASME PTB-3-2013 works its Part 5 examples on one vessel: an 84 in inside diameter shell with a 2:1 ellipsoidal top head and a forged 20 in top nozzle, 420 psig. PTB-3 models it with axisymmetric solid elements and reads stress classification lines. FEXLA models it as a shell. To separate the two, an independent axisymmetric solid of the same geometry (16,463 CAX8 elements in CalculiX, with the real 0.75 in and 1.5 in radii) was built and reproduces PTB-3 first.

PTB-3-2013 follows the 2013 edition of VIII-2. FEXLA follows the 2025 edition; where the rules differ (SPL, SPS, the fatigue equations), the comparison is of the stresses, and the report states the edition of each value.

The head and nozzle PL and the P+Q range are pinned in the test suite (head within 6 %, nozzle within 8 %; record T09). The P+Q range is read as the larger of the weld-toe value and the weld-line structural stress; chapter 9 explains why.

Before the toe reading was adopted, FEXLA read the junction checks at the mid-surface intersection, inside the 2 in forging wall, and with Tresca: PL came out 32 % high on the head and 67 % high on the nozzle. The PTB-3 runs found that, and the weld-toe reading fixed it. They are the reason the example stays in the test suite.

The limit-load route uses the layered MITC4 shell with J2 plasticity, small displacements, load control and Newton's method. The limit is the last converged load factor; the load step is cut down to 1e-4 of the reference load, so the limit is bracketed to that, or, where the step guard stops the run first, is a lower bound.

The clamped plate converges slowly with the size of the elements at its rim: +7.8 % on the coarsest full-plate mesh, +2.95 % at size factor 0.7. The 0.12 % above theory in B1, B2 and B7 on 64-facet rings is the faceting of the mesh: a ring of n flat elements carries its hoop force at cos(π/n).

F6 is outside its band against CalculiX's shells. CalculiX joins shells that meet at an angle with a rigid "knot" through the thickness, which moves the hinge of the collapse mechanism from the 8.18 mm nozzle wall into the 20 mm plate. A yield-line estimate with the hinge in the nozzle wall, the same plate and nozzle as an axisymmetric solid with a true T-joint, and a FEXLA run with the knot imitated all point the same way: FEXLA is 2.9 to 3.8 % below the solid, on the safe side. The report asks an engineer to accept the solid as F6's reference; until then it is listed open.

The cylinder's 10 % band is wide on purpose: the classical value has many half-waves, and a coarse mesh resolves them only approximately. The external-pressure buckling load includes the stiffness of pressure that follows the wall; before it was added, the ring case came out n²/(n² − 1) too high (record D01).

Buckling of the skirt access opening job has no independent cross-check yet, and there is no code_aster or MYSTRAN column for buckling (chapter 15).

A shell model carries no weld and no forging blend. At PTB-3's weld B (head-side toe of the nozzle forging) the shell's surface stress at the toe was 26 % below the solid's structural stress, which would over-predict fatigue life about 2.5×. Thicker bands, symmetric or eccentric, were tried and rejected: each one that raised the toe bending also lowered PL 10 to 15 % below the solid. What works is the equilibrium linearisation on the intersection line, from the element nodal forces: the mesh-insensitive structural stress of VIII-2 5.5.4.4.

Rule: primary checks (Pm, PL, PL + Pb, local failure) are read at the weld toe; the P + Q range is the larger of the toe and the weld-line value. The −11 % on damage follows from a 3 % lower structural stress raised to the curve's exponent (N ∝ ΔSess^−3.13).

Ratcheting (Bree) has no external cross-check. Its coverage rule (which fatigue event covers which row) is an engineering reading still to be confirmed (chapter 15).

On regular meshes FEXLA's brick is identical to CalculiX's (displacement to five digits, centroid stresses to 0.03 %). On the distorted block the two differ by 0.5 % in displacement: CalculiX's C3D8I uses a larger set of incompatible modes than Taylor's nine.

The brick submodel of a nozzle at a head's centre is driven at its cut faces by the shell solution. Its stress classification lines are linearised per VIII-2 Annex 5-A. On the default agitator nozzle (2:1 head ID 2000 × 14, nozzle 406.4 × 12.7), with and without a 700 × 14 pad, under 1 MPa and an 8 kN·m bending moment:

The remaining difference sits on the one line that starts and ends at the sharp corners where the neck meets the head (no fillet weld modelled). The peak stress at those corners is a singularity and is not compared. Whether the default submodel should be finer, or the table should warn on a line that ends at a sharp corner, is open item Q14.

A shell model of a nozzle is a cut-out of the vessel, held at its cut ends. If a cut is too close, it holds the shell round where the real vessel would let it oval. The study ran 16 radial nozzles (D/t 50 to 500, d/D 0.1 to 0.8) under pressure, radial force P, longitudinal moment ML and circumferential moment MC alone, with the shell 4 to 100 √(Rm·t) long on each side.

Measured against a model 100√(Rm·t) long, 50√(Rm·t) each side brings every load within 5 % for D/t up to 100 (worst 4.7 %, under P). On thinner shells P and MC keep changing out to the ovalisation length Rm·√(Rm/t): at 50√(Rm·t), MC is 7.6 % off at D/t 200 and 14.3 % at D/t 500, P 24 % and 31 %. There the real heads, rings or cone junctions decide, and the tool asks for the real span. The longer model costs elements: the default model grows from 6,480 to 16,932.

A far-field check now reads the hoop membrane at 2.5√(Rm·t) beyond the junction and flags a model more than 5 % off p·Rm/t. The study found the opening itself lowers that value by 1 to 3 % for d/D 0.1 and up to 17 % for d/D 0.8 at D/t 500, so a flag next to a large opening is not always an error.

The new length, the nozzle end choice and the far-field check come with the FEA workbench's 2026-11-01 release. Until that release is live, the workbench cuts the shell at max(4√(Rm·t), d, Rm) each side.

PL changes 0.5 % and PL + Pb + Q 0.14 % from 7.6k to 28k elements. One reading converges more slowly: the shell-side weld toe at the crotch under a circumferential moment, on a 1000 × 12 shell with a 219.1 × 8.18 nozzle. The adaptive mesh check refines where the recovery spread says the mesh is weakest, and repeats until the governing readings change less than 1 %.

The adaptive check adds 2,688 elements where the whole-mesh rerun adds 7,112, and lands within 1 % of the fine mesh; the test suite holds it to that (record T13). The governing pressure and moment readings stay within 1 % of a size factor 0.5 mesh.

Local refinement is built for a single nozzle on a cylinder, cone or flat plate. Elsewhere the check refines the whole mesh 0.7× and says so. The recovery spread of a four-node shell stays 6 to 25 % at junctions even on a converged mesh, so it drives where to refine but is not the stop condition. The adaptive check comes with the FEA workbench's 2026-11-01 release; before it, the check was one whole-mesh rerun at 0.7×.

Every run checks the shape of every element, the conditioning of the factorisation and the balance of loads and reactions. A model that fails a shape limit or is nearly singular is refused, not reported.

Why a structured mesh at the junction. The same nozzle model was meshed independently in Gmsh with the same size rule and analysed by the same solver. Where Gmsh's mesh is good, the global response agrees within 0.5 %. At the weld line its unstructured quads degenerate, exactly where the code check reads stresses:

The same comparison found a defect in FEXLA's own mesher: below size factor 0.5 the four corners of the O-grid folded. The shape check caught it and refused the model, so no wrong result reached a user. A local smoothing pass fixed it the same day: the nozzle now meshes validly down to size factor 0.3 (worst scaled Jacobian 0.77), and the default answer did not change (PL + Pb + Q 680.74 against 680.73 MPa).

The WRC 537 engine behind the calculator is recomputed on every build against the same geometry, loads and pressure run in commercial design software. A result that moves more than 0.3 % from the reference, or past a documented exception, fails the build. The table below is computed from that matrix when this page is built.

Every deviation above 0.3 % sits in the pressure stress intensity factor I of VIII-2 4.5, not in WRC 537. The full list, case by case, is on the validation page.

The rest of that published example differs: it prints one radial-load membrane value for all eight points, and derives its stress factors from the fillet radius. Without the bulletin's curve data in the repository the remaining gap was not resolved; it is open question Q-034.

WRC 537 against FEXLA's own FEA. Five round nozzles on cylinders (no pressure, loads at the junction, Kn = Kb = 1) were solved by both tools. The ratio is WRC 537 over FEA:

Under a radial load WRC 537 is above FEA in every case (1.02 to 1.50). Under moments it is up to 40 % below, the worst for the thick shell with a thin nozzle (G5, t/T 0.36), the known weak case of the bulletin method. With ML and MC together the FEA peak lies between WRC 537's four points, so the gap grows (0.69 for G1 with all six loads). The engine matches the reference software here; it is the method that differs. This is why the calculator offers the same case as a shell model, and why five nozzles are not yet a comparison over the bulletin's curve range (chapter 15).

Every item below is either a difference that is not explained to our satisfaction, a check that has not been run, or a choice an engineer still has to confirm. They are published so a reviewer can weigh them.