Numerical Methods
The layer between a formulation and a number. These articles cover sparse factorisation and ordering, Newton and arc-length iteration schemes, quadrature choices, and the conditioning and floating-point questions that decide whether two runs of the same model agree.
12 articles
Sphere tracing: the step that must not overshoot
A distance field that overestimates distance makes sphere tracing unsafe. In this tunnel the gradient bound is 1.1833, so only 0.8451 of each step is safe
The consistent tangent: 7 iterations against 887
Newton converges quadratically only with the exact Jacobian. On one elastoplastic bar the correct tangent needs 7 iterations where the elastic one needs 887.
Hoek-Brown tunnel: the closed-form ground reaction curve
A circular tunnel in Hoek-Brown rock, derived in closed form from radial equilibrium and solved in KATAI 2D: plastic radius, wall closure, every number given.
Reproducing a PLAXIS 2D benchmark to within 0.03%
A rigid strip footing on an elastic layer, solved in KATAI 2D against the published PLAXIS 2D value: every input, formula and convergence band stated.
Drucker-Prager: four ways to match Mohr-Coulomb
A cone fitted to a hexagon is right in some directions and wrong in others. Matched at triaxial compression it overstates extension strength by exactly 40%.
Grid Convergence Index: an error bar on a FEM result
Three meshes turn a finite element answer into an answer plus an uncertainty — observed order 1.8171, extrapolated 15.2067, fine-grid band 0.27%.
Sparse direct solvers: ordering, fill-in and the cost
The same matrix, factorised in two different orders, produces 498501 fill entries or none at all. Ordering is not a tuning parameter — it is most of the cost.
Catenary mooring: chain length, tension and anchor load
A catenary anchor sees pure horizontal load while chain rests on the seabed — 412.3 m of it here, 1350 kN of tension at the top, 1200 kN at the anchor.
Mohr-Coulomb return mapping: the corners and the apex
The single-surface return is three lines of algebra and it produces an inadmissible state. Worked in full, with the principal ordering violated by 37.494 kPa.
Gauss quadrature in finite elements: how many points
An n-point Gauss rule is exact to degree 2n-1, which decides everything. Under-integrate and the element develops modes that deform at no energy cost at all.
Plane strain vs axisymmetric: the depth the stress reaches
A strip and a circle of the same width put very different loads on the ground below. The 10% contour sits at 1.85B under one and 6.34B under the other.
Terzaghi consolidation: the series and its approximations
The two formulas every textbook gives for the degree of consolidation are approximations to one Fourier series. Here is the series, and where each one fails.
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