// CPU reference implementation of the spatial discretization (float64).
// It mirrors step.wgsl exactly and is used for the stable time-step estimate
// and for the tests. It is not used in the time loop of the browser simulation.

import { type Grid, type Material, trapWeight } from "../params.ts";

export interface Fields {
  u: Float64Array;
  v: Float64Array;
  w: Float64Array;
}

const G3 = 1 / Math.sqrt(3);

/**
 * Accelerations (au, av, aw) of every node for displacement fields (u, v, w),
 * without damping. Bending forces come from the discrete Kirchhoff energy
 * (natural free-edge conditions); membrane forces come from bilinear (Q4)
 * elements with 2x2 Gauss integration and the von Karman strains.
 */
export function accelerations(g: Grid, m: Material, f: Fields, out: Fields, w0?: Float64Array): void {
  const { nx, ny, hx, hy } = g;
  const { u, v, w } = f;
  const hx2 = hx * hx;
  const hy2 = hy * hy;
  const nu = m.nu;
  const id = (i: number, j: number) => j * nx + i;

  // Bending moment-like quantities at nodes (zero where the second difference is undefined).
  const Ma = (i: number, j: number): number => {
    if (i < 1 || i > nx - 2 || j < 0 || j > ny - 1) return 0;
    const a = (w[id(i + 1, j)] - 2 * w[id(i, j)] + w[id(i - 1, j)]) / hx2;
    let b = 0;
    if (j >= 1 && j <= ny - 2) b = (w[id(i, j + 1)] - 2 * w[id(i, j)] + w[id(i, j - 1)]) / hy2;
    return trapWeight(g, i, j) * (a + nu * b);
  };
  const Mb = (i: number, j: number): number => {
    if (j < 1 || j > ny - 2 || i < 0 || i > nx - 1) return 0;
    const b = (w[id(i, j + 1)] - 2 * w[id(i, j)] + w[id(i, j - 1)]) / hy2;
    let a = 0;
    if (i >= 1 && i <= nx - 2) a = (w[id(i + 1, j)] - 2 * w[id(i, j)] + w[id(i - 1, j)]) / hx2;
    return trapWeight(g, i, j) * (b + nu * a);
  };
  // Mixed second difference at a cell center (zero for cells outside the plate).
  const Cc = (ci: number, cj: number): number => {
    if (ci < 0 || ci > nx - 2 || cj < 0 || cj > ny - 2) return 0;
    return (w[id(ci + 1, cj + 1)] - w[id(ci + 1, cj)] - w[id(ci, cj + 1)] + w[id(ci, cj)]) / (hx * hy);
  };

  const q = (hx * hy) / 4; // area per Gauss point
  const U = new Float64Array(4);
  const V = new Float64Array(4);
  const Wn = new Float64Array(4);
  const W0 = new Float64Array(4);
  const dNdx = new Float64Array(4);
  const dNdy = new Float64Array(4);

  // Membrane force on local node `a` of cell (ci, cj); returns [Fu, Fv, Fw].
  const cellForce = (ci: number, cj: number, a: number, F: Float64Array): void => {
    if (ci < 0 || ci > nx - 2 || cj < 0 || cj > ny - 2) return;
    const n0 = id(ci, cj), n1 = id(ci + 1, cj), n2 = id(ci + 1, cj + 1), n3 = id(ci, cj + 1);
    U[0] = u[n0]; U[1] = u[n1]; U[2] = u[n2]; U[3] = u[n3];
    V[0] = v[n0]; V[1] = v[n1]; V[2] = v[n2]; V[3] = v[n3];
    Wn[0] = w[n0]; Wn[1] = w[n1]; Wn[2] = w[n2]; Wn[3] = w[n3];
    if (w0) { W0[0] = w0[n0]; W0[1] = w0[n1]; W0[2] = w0[n2]; W0[3] = w0[n3]; }
    for (let gp = 0; gp < 4; gp++) {
      const xi = gp & 1 ? G3 : -G3;
      const eta = gp & 2 ? G3 : -G3;
      dNdx[0] = (-(1 - eta) / 4) * (2 / hx);
      dNdx[1] = ((1 - eta) / 4) * (2 / hx);
      dNdx[2] = ((1 + eta) / 4) * (2 / hx);
      dNdx[3] = (-(1 + eta) / 4) * (2 / hx);
      dNdy[0] = (-(1 - xi) / 4) * (2 / hy);
      dNdy[1] = (-(1 + xi) / 4) * (2 / hy);
      dNdy[2] = ((1 + xi) / 4) * (2 / hy);
      dNdy[3] = ((1 - xi) / 4) * (2 / hy);
      // Marguerre strains: total slope (w0 + w) minus the stress-free slope of w0.
      let ux = 0, uy = 0, vx = 0, vy = 0, wx = 0, wy = 0, w0x = 0, w0y = 0;
      for (let b = 0; b < 4; b++) {
        ux += dNdx[b] * U[b]; uy += dNdy[b] * U[b];
        vx += dNdx[b] * V[b]; vy += dNdy[b] * V[b];
        wx += dNdx[b] * (Wn[b] + W0[b]); wy += dNdy[b] * (Wn[b] + W0[b]);
        w0x += dNdx[b] * W0[b]; w0y += dNdy[b] * W0[b];
      }
      const exx = ux + 0.5 * (wx * wx - w0x * w0x);
      const eyy = vy + 0.5 * (wy * wy - w0y * w0y);
      const gxy = uy + vx + wx * wy - w0x * w0y;
      const Nxx = m.C * (exx + nu * eyy);
      const Nyy = m.C * (eyy + nu * exx);
      const Nxy = m.G * gxy;
      F[0] -= q * (Nxx * dNdx[a] + Nxy * dNdy[a]);
      F[1] -= q * (Nyy * dNdy[a] + Nxy * dNdx[a]);
      F[2] -= q * ((Nxx * wx + Nxy * wy) * dNdx[a] + (Nxy * wx + Nyy * wy) * dNdy[a]);
    }
  };

  const F = new Float64Array(3);
  for (let j = 0; j < ny; j++) {
    for (let i = 0; i < nx; i++) {
      const k = id(i, j);
      const bend =
        (Ma(i - 1, j) - 2 * Ma(i, j) + Ma(i + 1, j)) / hx2 +
        (Mb(i, j - 1) - 2 * Mb(i, j) + Mb(i, j + 1)) / hy2 +
        (2 * (1 - nu) * (Cc(i, j) - Cc(i - 1, j) - Cc(i, j - 1) + Cc(i - 1, j - 1))) / (hx * hy);
      F[0] = 0; F[1] = 0; F[2] = -m.D * hx * hy * bend;
      cellForce(i, j, 0, F);
      cellForce(i - 1, j, 1, F);
      cellForce(i - 1, j - 1, 2, F);
      cellForce(i, j - 1, 3, F);
      const mass = m.rhoh * hx * hy * trapWeight(g, i, j);
      out.u[k] = F[0] / mass;
      out.v[k] = F[1] / mass;
      out.w[k] = F[2] / mass;
    }
  }
}

/**
 * Largest angular frequency of the linearized discrete system, by power
 * iteration on a small grid with the same spacing (the extreme mode is a
 * grid-scale mode, so the plate size does not matter).
 */
export function maxAngularFrequency(g: Grid, m: Material, nSmall = 12, iters = 300): number {
  const gs: Grid = { nx: nSmall, ny: nSmall, hx: g.hx, hy: g.hy, n: nSmall * nSmall };
  const n = gs.n;
  const x: Fields = { u: new Float64Array(n), v: new Float64Array(n), w: new Float64Array(n) };
  const y: Fields = { u: new Float64Array(n), v: new Float64Array(n), w: new Float64Array(n) };
  let seed = 12345;
  const rnd = () => {
    seed = (seed * 1664525 + 1013904223) >>> 0;
    return seed / 4294967296 - 0.5;
  };
  for (let k = 0; k < n; k++) {
    x.u[k] = rnd(); x.v[k] = rnd(); x.w[k] = rnd();
  }
  const eps = 1e-9; // keeps the nonlinear terms negligible
  let lambda = 0;
  for (let it = 0; it < iters; it++) {
    let norm = 0;
    for (let k = 0; k < n; k++) norm += x.u[k] ** 2 + x.v[k] ** 2 + x.w[k] ** 2;
    norm = Math.sqrt(norm);
    for (let k = 0; k < n; k++) {
      x.u[k] *= eps / norm; x.v[k] *= eps / norm; x.w[k] *= eps / norm;
    }
    accelerations(gs, m, x, y);
    let dot = 0;
    for (let k = 0; k < n; k++) dot += x.u[k] * y.u[k] + x.v[k] * y.v[k] + x.w[k] * y.w[k];
    lambda = -dot / (eps * eps); // Rayleigh quotient of -M^{-1}K
    for (let k = 0; k < n; k++) {
      x.u[k] = -y.u[k]; x.v[k] = -y.v[k]; x.w[k] = -y.w[k];
    }
  }
  return Math.sqrt(Math.max(lambda, 0));
}

/** Discrete potential energy (bending + membrane) matching `accelerations`. */
export function potentialEnergy(g: Grid, m: Material, f: Fields, w0?: Float64Array): number {
  const { nx, ny, hx, hy } = g;
  const { u, v, w } = f;
  const nu = m.nu;
  const id = (i: number, j: number) => j * nx + i;
  let Ub = 0;
  for (let j = 0; j < ny; j++) {
    for (let i = 0; i < nx; i++) {
      let a = 0, b = 0;
      if (i >= 1 && i <= nx - 2) a = (w[id(i + 1, j)] - 2 * w[id(i, j)] + w[id(i - 1, j)]) / (hx * hx);
      if (j >= 1 && j <= ny - 2) b = (w[id(i, j + 1)] - 2 * w[id(i, j)] + w[id(i, j - 1)]) / (hy * hy);
      Ub += trapWeight(g, i, j) * (a * a + b * b + 2 * nu * a * b);
    }
  }
  for (let cj = 0; cj < ny - 1; cj++) {
    for (let ci = 0; ci < nx - 1; ci++) {
      const c = (w[id(ci + 1, cj + 1)] - w[id(ci + 1, cj)] - w[id(ci, cj + 1)] + w[id(ci, cj)]) / (hx * hy);
      Ub += 2 * (1 - nu) * c * c;
    }
  }
  Ub *= 0.5 * m.D * hx * hy;

  let Um = 0;
  const q = (hx * hy) / 4;
  for (let cj = 0; cj < ny - 1; cj++) {
    for (let ci = 0; ci < nx - 1; ci++) {
      const nn = [id(ci, cj), id(ci + 1, cj), id(ci + 1, cj + 1), id(ci, cj + 1)];
      for (let gp = 0; gp < 4; gp++) {
        const xi = gp & 1 ? G3 : -G3;
        const eta = gp & 2 ? G3 : -G3;
        const dNdx = [-(1 - eta), 1 - eta, 1 + eta, -(1 + eta)].map((s) => (s / 4) * (2 / hx));
        const dNdy = [-(1 - xi), -(1 + xi), 1 + xi, 1 - xi].map((s) => (s / 4) * (2 / hy));
        let ux = 0, uy = 0, vx = 0, vy = 0, wx = 0, wy = 0, w0x = 0, w0y = 0;
        for (let b = 0; b < 4; b++) {
          const z0 = w0 ? w0[nn[b]] : 0;
          ux += dNdx[b] * u[nn[b]]; uy += dNdy[b] * u[nn[b]];
          vx += dNdx[b] * v[nn[b]]; vy += dNdy[b] * v[nn[b]];
          wx += dNdx[b] * (w[nn[b]] + z0); wy += dNdy[b] * (w[nn[b]] + z0);
          w0x += dNdx[b] * z0; w0y += dNdy[b] * z0;
        }
        const exx = ux + 0.5 * (wx * wx - w0x * w0x);
        const eyy = vy + 0.5 * (wy * wy - w0y * w0y);
        const gxy = uy + vx + wx * wy - w0x * w0y;
        Um += q * 0.5 * (m.C * (exx * exx + eyy * eyy + 2 * nu * exx * eyy) + m.G * gxy * gxy);
      }
    }
  }
  return Ub + Um;
}