Verification & Validation
A commercial result is only worth what you can defend. This page is the public verification record of KATAI 2D: each row is a closed-form solution, a canonical benchmark or a documented PLAXIS 2D result, reproduced to a stated tolerance by an automated test that runs on every build. Nothing here is hand-picked or hand-tuned.
How we verify
We follow the established verification-and-validation discipline (Roache): build up from the problems with an exact closed-form answer — elastic fields, earth pressures, limit loads — before trusting the harder, coupled and nonlinear cases.
Every capability is locked to its mathematical source — a journal paper, a standard monograph, or the published theory of the established codes — in the Scientific Manual, and then checked against that source automatically. Where a result is sensitive to mesh or to a known limitation of the theory, we say so on this page, in the same place as the agreement.
01 · Plasticity & limit loads
Bearing capacity, earth pressure and slope stability — the loads at which the soil fails, against the classical limit solutions.
| Benchmark | Reference | KATAI · error | Automated test |
|---|---|---|---|
| Prandtl strip-footing bearing factor Nc (φ = 0) | 2 + π = 5.14 | 5.20 · 1.1% | test_tri15_prandtl |
| Reissner bearing factor Nc (φ = 20°) | (Nq−1)cotφ = 14.84 | 15.13 · 2.0% | test_bearing_phi |
| Thick-cylinder collapse pressure (axisymmetric) | 2c·ln(b/a) | 0.3% | test_axisym_collapse |
| Rankine active thrust | ½KaγH² = 300 | 300.0 · round-off | test_earth_pressure |
| Rankine passive thrust | ½KpγH² = 2700 | 2700 · <0.1% | test_earth_pressure |
| Rankine at-rest thrust (K0) | ½K0γH² = 450 | 450.0 · round-off | test_earth_pressure |
| Slope factor of safety (φ‑c reduction) | Bishop / Griffiths & Lane ≈ 0.99 | 1.01 · 2.0% | test_slope |
02 · Linear-elastic closed form
Where an analytical field exists, the finite element solution should recover it to round-off — and it does.
| Benchmark | Reference | Result | Automated test |
|---|---|---|---|
| Lamé thick cylinder — σr, σθ (axisymmetric) | Lamé | 10−5 rel. | test_axisym_cylinder |
| Boussinesq strip load — σz | Boussinesq | <5% | test_boussinesq |
| Flamant line load — σz = 2P/πz | Flamant | <6% (edge) | test_flamant |
| Cantilever plate — PL³/3EI + PL/kGA′ | Timoshenko | round-off | test_plate, test_plate5 |
| Inverse isoparametric map (curved element) | round-trip identity | 1.4×10−15 | test_point_location |
03 · Groundwater flow & consolidation
Steady-state flow against the classical hydraulics, and time-dependent consolidation against the Terzaghi series.
| Benchmark | Reference | Result | Automated test |
|---|---|---|---|
| 1D Darcy column — head & discharge | analytical | round-off | test_seepage |
| Radial (Thiem) flow — head field | Thiem | order O(h³) | test_seepage |
| Radial (Thiem) flow — discharge | Thiem | 10−6 rel. | test_seepage |
| Dam-base uplift — h(x) = (H/π)arccos(x/b) | Harr | 0.49% | test_seepage |
| Dupuit / Charny two-reservoir discharge | Charny | 0.08% | test_seepage |
| Manufactured-solution convergence | Roache MMS | tri6 O(h³) · tri15 O(h⁵) | test_seepage |
| Terzaghi 1D consolidation — U(Tv) | Terzaghi series | <2% | test_consolidation |
| HS small stiffness degradation (Gs, Gt, cut-off) | Hardin–Drnevich | round-off | test_hssmall |
04 · Structural & interface elements
Plates, anchors, geogrids, embedded beams and Coulomb interfaces — each checked against its own closed-form or limit solution.
| Benchmark | Reference | Result | Automated test |
|---|---|---|---|
| Anchor (fixed-end + node-to-node), elastoplastic | closed-form | round-off | test_anchor_plastic |
| Geogrid — tension-only + plastic cap Np | closed-form | round-off | test_geogrid |
| Coulomb interface — collapse multiplier λ* | limit analysis | 0.25–1% | test_interface |
| Interface K0 install (wished-in-place) — max |u| | self-equilibrium = 0 | 4×10−12 | test_wall_k0_excavation |
| Embedded beam — axial, constant soil | (P/EAλ)coth(λL) | round-off | test_embedded_beam |
| Embedded beam — load transfer Σ skin | Σ skin = P | round-off | test_embedded_beam |
| Embedded beam — ultimate capacity | Qskin + Qbase | −0.1% | test_embedded_beam |
05 · PLAXIS 2D benchmarks
Beyond the analytical checks: full, integrated models reproduced from documented PLAXIS 2D and peer-reviewed sources — finite element against finite element.
PLAXIS 2D Tutorial · Lesson 1
Axisymmetric, Mohr‑Coulomb (drained), K0 initial stress with a water table, the rigid footing pushed to 0.05 m — near the bearing capacity. 640 elements, 2 551 DOFs, reproduced from scratch.
+3.2% total footing reaction at 0.05 m settlement
Paul, Halder & Mukherjee · IJCRT 2024
Purely cohesive clay, single-stage excavation, modelled with the rigorous undrained effective-stress method (νu → 0.5). Maximum bending moment for the well-posed case (Cu=30, H=5.0 m).
Honest caveat: deep walls converge toward PLAXIS as the model boundary recedes from the failure wedge (a setup effect, not the physics); near-critical embedments are ill-conditioned — PLAXIS itself failed to converge on the adjacent cases.
06 · The "Calculate" path
The checks above exercise the numerical kernels directly. These verify the exact path a user drives — geometry → mesh → assembly → solve → stress recovery — when they press Calculate.
| Benchmark | Reference | KATAI (GUI) · error | Automated test |
|---|---|---|---|
| 1D oedometer (self-weight + surcharge) via "Calculate" | −γH²/2Eoed, −qH/Eoed | <2% | test_solve |
| K0 procedure — max |u| | geostatic equilibrium = 0 | 10−13 | test_gui_solve · diag_k0 |
| K0 effective σ′v, σ′h = K0σ′v | −γ(H−y) | 10−13 | diag_k0 |
| Axisymmetric gravity oedometer — −γH²/2Eoed | −0.066857 | −0.066859 · <0.01% | test_axisym_gui |
| Axisymmetric radial expansion — ur(R) = νqR/E | 0.003000 | 0.003000 · round-off | test_axisym_gui |
| Systematic K0 admissibility (LE/MC × tri6/tri15 × 5 structures) | geostatic, max |u| = 0 | 20/20 | test_gui_matrix |
| MC Prandtl Nc (φ = 0) through the GUI path | 5.14 | 4.84 · −5.9% (conservative) | study_gui_validation |
| Consolidation — U(Tv = 0.6) through the GUI | Terzaghi 0.8156 | 0.8100 · −0.7% | test_consolidation_gui |
| Consolidation — final settlement | 0.11930 m | 0.11923 m · 0.06% | test_consolidation_gui |
| Undrained (B) strip footing qult (φ′=30 forced to 0) | (2+π)·su = 5.142 | 5.175 su · +0.6% | study_gui_validation |
| Safety (φ‑c reduction) slope FoS through the GUI | Bishop ≈ 0.99 | 1.009 · 1.9% | test_safety_gui |
The Mohr‑Coulomb collapse load through the GUI is ~6% below the exact limit — the conservative, safe-side effect of the incremental load-step granularity, not a modelling error. The elastic and service response is exact, and the dedicated solver path reaches the limit to ~1%.
07 · Speed & robustness
A sparse direct core with pattern-aware factorisation and parallel assembly — and a contract that it never crashes on a broken model.
| Scenario | Size | Time |
|---|---|---|
| Linear-elastic gravity | 48 266 DOFs | 0.35 s |
| Mohr‑Coulomb footing, pushed to collapse | 5 178 DOFs | 5.9 s |
| Hardening Soil footing, service load (40 steps, full convergence) | 2 892 DOFs | 16.1 s |
2.7–2.9× faster than the un-optimised baseline, bit-for-bit identical results — measured on a 2017 laptop CPU (4 cores).
adversarial, degenerate and extreme inputs survived — zero crashes, zero NaN.
Empty and contradictory materials, zero-area and self-intersecting geometry, E = 0 / ∞ / NaN, singular and rigid-body systems, out-of-domain structures, and a full complex valid model — all return an honest result or a clear message, never a crash. This contract is itself a locked regression test.
test_robustness
The complete theoretical and numerical basis behind every number on this page is documented in the Scientific Manual, released for academic and professional review with early access.