import type { MethodDefinition } from "./types";
import { GOVERNING_EQUATIONS, RADIATION_EQUATIONS, STRESS_EQUATIONS } from "./common";

export const MODAL_METHOD: MethodDefinition = {
  id: "modal",
  name: "Modal superposition",
  shortName: "Modal",
  summary: "Closed-form modes of the simply supported rectangle, each integrated exactly in time.",
  solverFunction: "plate_modal",
  supports: {
    shape: ["rectangle"],
    bc: ["simply-supported"],
    excitation: ["impulse", "harmonic"],
    audio: "full",
  },
  readingOrder: [
    "plate_modal.m",
    "plate_setup.m",
    "plate_force_time.m",
    "plate_force_footprint.m",
    "plate_sdof.m",
    "plate_stress.m",
    "plate_listener.m",
    "plate_spectrum.m",
    "plate_result_schema.m",
  ],
  equations: [
    ...GOVERNING_EQUATIONS,
    {
      tex: String.raw`\phi_{mn}(x,y) = \sin\frac{m\pi x}{a}\,\sin\frac{n\pi y}{b}, \qquad \omega_{mn} = \kappa\left[\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2\right]`,
      caption:
        "Eigenfunctions and natural frequencies of the simply supported rectangle; exact for w = 0 and w_nn = 0 on all four edges.",
    },
    {
      tex: String.raw`w = \sum_{mn} q_{mn}(t)\,\phi_{mn}(x,y), \qquad \ddot q_{mn} + 2\sigma_{mn}\dot q_{mn} + \omega_{mn}^2 q_{mn} = \frac{G_{mn}F(t)}{M_{mn}}`,
      caption:
        "Each mode is an independent damped oscillator with modal mass M = mu a b / 4, force factor G = integral of s phi, and damping rate sigma = sigma0 + sigma1 lambda.",
    },
    {
      tex: String.raw`\begin{pmatrix} q_{n+1} \\ \dot q_{n+1} \end{pmatrix} = \mathbf A(\Delta t) \begin{pmatrix} q_n \\ \dot q_n \end{pmatrix} + \mathbf B(\Delta t) \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix}`,
      caption:
        "Exact state transition over one sample for a force that is linear between samples (Nigam and Jennings 1969). No stability limit and no phase error.",
    },
    ...STRESS_EQUATIONS,
    ...RADIATION_EQUATIONS,
  ],
  validity: [
    "Exact only for the simply supported rectangle; other edge conditions and the ellipse have no closed-form modes of this kind (a disk would need Bessel functions).",
    "The modal sum is truncated at fmax and max_modes. Displacement converges quickly with mode count; stress converges more slowly, most of all near the strike point.",
    "The footprint projection and the Rayleigh integral are evaluated by quadrature on fixed grids.",
    "Damping rates are a model with two parameters, chosen, not measured.",
  ],
  durationOf: (spec) => spec.output.duration,
  settingsKey: "modal",
  usPerCellStep: 0,
};