KATAI 2D - Scientific Manual

KATAI 2D is an open-formulation, verification-driven finite element platform for two-dimensional geotechnical analysis under plane-strain and axisymmetric conditions. This Scientific Manual documents its theoretical and numerical foundations and the verification evidence behind every capability. The manual covers: the small-strain continuum boundary-value problem and its Galerkin discretisation; the six- and fifteen-node triangular element family; the constitutive library — linear elasticity, the Mohr–Coulomb model with an exact principal-stress return mapping, and the Hardening Soil model with optional small-strain stiffness; a globally convergent Newton–Raphson solver with a consistent (algorithmic) tangent, line search and adaptive load stepping; steady-state confined and unconfined groundwater flow with seepage faces; Biot consolidation with an elastoplastic skeleton; the structural and interface elements (plate, anchor, geogrid, embedded beam, zero-thickness interface); and the analysis types of routine practice — initial K0 stress, gravity loading, plastic, strength-reduction (safety), time-dependent consolidation and multi-phase staged construction. The guiding principle of the project is that every capability is locked to its mathematical source — a primary reference, a standard monograph, or an authoritative formulation from the literature — and then verified against a closed-form or canonical benchmark before it is considered complete. The implementation reproduces the classical analytical solutions of geotechnical engineering to within engineering tolerance: the Prandtl bearing-capacity factor Nc = 2 + π to 1.1%, the Lamé thick-cylinder stress field to a relative error of 1e-5, the one-dimensional Terzaghi consolidation curve to below 2%, the Thiem radial-flow discharge to a relative error of 1e-6, and the manufactured-solution convergence rates of the seepage operator at the theoretical orders O(h³) for the quadratic element and O(h⁵) for the quartic element. Every numerical claim is backed by an automated test that runs on every build. This document corresponds to KATAI 2D version 0.2. Capabilities that are formulated but not yet released are clearly marked. It is released, together with the verification suite it describes, for academic and professional review.