Summary

• Adds a 1D Euler-Bernoulli beam bending solver (eulerBernoulliBeamScript) using cubic
  Hermite (C¹) finite elements, so both deflection w and slope theta = dw/dx are
  represented as nodal DOFs. The fourth-order analogue of the existing scalar 1D solvers
  (heatConductionScript, generalFormPDEScript).
• Adds hermiteCubic support to basisFunctions.js (shape functions + 2nd ksi-derivative)
  and a matching 6-point Gauss rule to numericalIntegration.js. This is general C¹
  interpolation infrastructure, not beam-specific, eulerBernoulliBeam.js is just its
  first consumer, and it should be directly reusable for future 4th-order models (e.g.
  Kirchhoff plate bending).
• Adds a worked example reproducing the "Bending of a Beam" problem from J.N. Reddy,
  An Introduction to the Finite Element Method, 3rd ed. (FEM1D reference program,
  Chapter 7): a beam clamped at one end, supported by a roller at midspan and a linear
  spring at the free end, carrying a distributed load, a point moment, and a point load.

What's included

• src/mesh/basisFunctions.js — cubic Hermite basis functions and their 1st/2nd
  ksi-derivatives, on the same [0,1] natural-coordinate convention used by the existing
  linear/quadratic Lagrange elements in the library.
• src/methods/numericalIntegration.js — 6-point Gauss quadrature rule for hermiteCubic
  (exact up to degree 11).
• src/models/eulerBernoulliBeam.js (new) — element assembly
  (assembleEulerBernoulliBeamMat): builds the global stiffness/load system from
  EI(x), c0(x) (elastic foundation, optional), and q(x) (distributed load).
• src/models/beamBoundaryConditions.js (new) — essential (fixed/pinned/rotation),
  natural (force/moment), and mixed/Robin (spring) boundary conditions. Unlike the
  side-based "0"/"1" boundary keys used elsewhere, these are keyed by 1-based node
  number, since beam problems commonly need supports/loads at interior nodes (e.g. a
  midspan roller), not just the two domain ends.
• src/FEAScript.js — wires up solverConfig: "eulerBernoulliBeamScript".
• examples/Beam1DFEM/Beam1DEuler_Bernoulli.js + README.md — the worked example above,
  plus documentation of the mesh/coefficient/boundary-condition API for this solver.
• tests/unit/eulerBernoulliBeam.test.js — verifies the single-element stiffness matrix
  against the closed-form textbook Hermite beam matrix (2 EI/L combinations), and a
  cantilever tip-load case against the classical closed-form solution
  (w = PL³/3EI, theta = PL²/2EI).
• tests/regression/EulerBernoulliBeam/ — regression baseline for the full worked example,
  plus an independent global force/moment equilibrium check (recomputed reactions from
  the raw, unconstrained assembly) that holds regardless of the specific baseline numbers.

Testing

• npm test passes (5/5 test files, including the 2 new ones)
• Single-element stiffness matrix matches the closed-form Hermite beam matrix to ~1e-10
• Cantilever tip deflection/rotation matches the classical closed-form solution exactly
• Full worked-example solution satisfies global force and moment equilibrium to ~1e-11
• npm run build succeeds

Notes for reviewers

• The beam geometry mesh only needs elementOrder: "linear" (2 end nodes per element);
  the cubic Hermite field interpolation is applied internally regardless of this setting,
  a standard "subparametric" formulation for beam elements.
• See examples/Beam1DFEM/README.md for the boundary-condition syntax and a table of
  supported condition types.