// One explicit time step (symplectic Euler) of the von Karman plate with a
// thickness field.
//
// State layout (structure of arrays, n = nx * ny nodes):
//   pos: [u | v | w | wLo]   wLo carries the rounding error of the w update
//   vel: [du | dv | dw]
// A workgroup of 16x16 threads advances a 12x12 tile of nodes in three
// phases separated by barriers. (1) Every thread loads one entry of the tile
// plus its 2-node halo (16x16 values of w, u, v, w0, the thickness, and the
// node weights) into shared memory. (2) The threads compute, once each
// and again into shared memory, the membrane forces of the 13x13 cells
// around the tile and the weighted bending moments and mixed differences
// that the bending stencil gathers. (3) Each of the 144 tile nodes evaluates
// its update from shared memory alone. On grids of a few thousand nodes the
// kernel is bound by the latency of one workgroup rather than by throughput,
// so the shared phases, which replace a tenfold per-node recomputation of
// the stencil ingredients, are what make a step cheap.
//
// The thickness enters as the local bending stiffness D(h) at the nodes and
// the cells, the local membrane stiffnesses at the cells (mean thickness of
// the four corners), and the node mass. Nodes outside the plate (weight 0)
// are skipped: a second difference exists only where the node and both
// neighbors belong to the plate, and a cell contributes only when its four
// corners do. That reproduces the free edge of the rectangle and gives a
// staircase free edge for any other outline, including a center hole.

@group(0) @binding(0) var<uniform> P: Params;
@group(0) @binding(1) var<uniform> SP: StepParams;
@group(0) @binding(2) var<storage, read> posIn: array<f32>;
@group(0) @binding(3) var<storage, read> velIn: array<f32>;
@group(0) @binding(4) var<storage, read_write> posOut: array<f32>;
@group(0) @binding(5) var<storage, read_write> velOut: array<f32>;
@group(0) @binding(6) var<storage, read_write> hist: array<f32>; // [slot * n + node]: transverse velocity at the audio sample instants
// Two 4-float slots of drumstick state (position, velocity, force, max force).
@group(0) @binding(7) var<storage, read_write> stick: array<f32>;

const TILE = 12;  // nodes per workgroup along each axis (must match simulation.ts)
const HALO = 2;
const TW = 16;    // TILE + 2 * HALO: the loaded tile, one thread per entry
const CT = 13;    // TILE + 1: cells adjacent to the tile's nodes
const MT = 14;    // TILE + 2: nodes at which the bending moments are needed
// Tile with halo, local index (j - oy) * TW + (i - ox).
var<workgroup> tw: array<f32, 256>;  // w
var<workgroup> tu: array<f32, 256>;  // u
var<workgroup> tv: array<f32, 256>;  // v
var<workgroup> tw0: array<f32, 256>; // stress-free shape
var<workgroup> th: array<f32, 256>;  // thickness
var<workgroup> tn: array<u32, 64>;   // node weights in quarters (0..4), four to a word
// Membrane force contributions of the CT x CT cells, 4 nodes x 3 components each.
var<workgroup> cellF: array<f32, 2028>;
// Weighted bending moments at the MT x MT inner nodes and mixed differences at the CT x CT cells.
var<workgroup> ma: array<f32, 196>;
var<workgroup> mb: array<f32, 196>;
var<workgroup> cc: array<f32, 169>;

// Tile origin in global node coordinates (set once per invocation).
var<private> ox: i32;
var<private> oy: i32;
var<private> ihx2: f32;
var<private> ihy2: f32;
var<private> ihxy: f32;

fn id(i: i32, j: i32) -> u32 { return u32(j) * P.nx + u32(i); }
fn lidx(i: i32, j: i32) -> u32 { return u32(j - oy) * u32(TW) + u32(i - ox); }
fn midx(i: i32, j: i32) -> u32 { return u32(j - oy - 1) * u32(MT) + u32(i - ox - 1); }
fn cidx(ci: i32, cj: i32) -> u32 { return u32(cj - oy - 1) * u32(CT) + u32(ci - ox - 1); }
// Tile lookups with global indices; the tile covers [ox, ox + TW) x [oy, oy + TW).
fn W(i: i32, j: i32) -> f32 { return tw[lidx(i, j)]; }
fn U(i: i32, j: i32) -> f32 { return tu[lidx(i, j)]; }
fn V(i: i32, j: i32) -> f32 { return tv[lidx(i, j)]; }
fn W0(i: i32, j: i32) -> f32 { return tw0[lidx(i, j)]; }
// Transverse velocity, read from global memory (only the loss Laplacian needs it,
// and keeping it out of shared memory keeps the workgroup within 16 KB).
fn DW(i: i32, j: i32) -> f32 { return velIn[2u * P.n + id(i, j)]; }
fn TH(i: i32, j: i32) -> f32 { return th[lidx(i, j)]; }
fn NWq(i: i32, j: i32) -> u32 { let l = lidx(i, j); return (tn[l >> 2u] >> ((l & 3u) * 8u)) & 0xFFu; }
fn NW(i: i32, j: i32) -> f32 { return 0.25 * f32(NWq(i, j)); }
fn INS(i: i32, j: i32) -> bool { return NWq(i, j) != 0u; }
fn insCell(ci: i32, cj: i32) -> bool { return INS(ci, cj) && INS(ci + 1, cj) && INS(ci + 1, cj + 1) && INS(ci, cj + 1); }
// Bending stiffness at a node and at a cell (mean thickness of the four corners).
fn Dn(i: i32, j: i32) -> f32 { let h = TH(i, j); return P.Db * h * h * h; }
fn cellH(ci: i32, cj: i32) -> f32 { return 0.25 * (TH(ci, cj) + TH(ci + 1, cj) + TH(ci + 1, cj + 1) + TH(ci, cj + 1)); }

// Graph Laplacian of the transverse velocity at node (i, j), over neighbors in the plate.
fn lapDW(i: i32, j: i32) -> f32 {
  let c = DW(i, j);
  var lap = 0.0;
  if (INS(i - 1, j)) { lap += (DW(i - 1, j) - c) * ihx2; }
  if (INS(i + 1, j)) { lap += (DW(i + 1, j) - c) * ihx2; }
  if (INS(i, j - 1)) { lap += (DW(i, j - 1) - c) * ihy2; }
  if (INS(i, j + 1)) { lap += (DW(i, j + 1) - c) * ihy2; }
  return lap;
}

// Weighted bending moments at nodes; zero where the second difference is undefined.
fn Ma(i: i32, j: i32) -> f32 {
  if (!(INS(i - 1, j) && INS(i, j) && INS(i + 1, j))) { return 0.0; }
  let a = (W(i + 1, j) - 2.0 * W(i, j) + W(i - 1, j)) * ihx2;
  var b = 0.0;
  if (INS(i, j - 1) && INS(i, j + 1)) { b = (W(i, j + 1) - 2.0 * W(i, j) + W(i, j - 1)) * ihy2; }
  return NW(i, j) * Dn(i, j) * (a + P.nu * b);
}
fn Mb(i: i32, j: i32) -> f32 {
  if (!(INS(i, j - 1) && INS(i, j) && INS(i, j + 1))) { return 0.0; }
  let b = (W(i, j + 1) - 2.0 * W(i, j) + W(i, j - 1)) * ihy2;
  var a = 0.0;
  if (INS(i - 1, j) && INS(i + 1, j)) { a = (W(i + 1, j) - 2.0 * W(i, j) + W(i - 1, j)) * ihx2; }
  return NW(i, j) * Dn(i, j) * (b + P.nu * a);
}
// Weighted mixed second difference at a cell center; zero outside the plate.
fn Cc(ci: i32, cj: i32) -> f32 {
  if (!insCell(ci, cj)) { return 0.0; }
  let h = cellH(ci, cj);
  return P.Db * h * h * h * (W(ci + 1, cj + 1) - W(ci + 1, cj) - W(ci, cj + 1) + W(ci, cj)) * ihxy;
}

// Membrane forces of the bilinear cell (ci, cj) on its four nodes (2x2 Gauss),
// written to cellF at local cell index lc.
fn cellForces(ci: i32, cj: i32, lc: u32) {
  let base = lc * 12u;
  if (!insCell(ci, cj)) {
    for (var q = 0u; q < 12u; q++) { cellF[base + q] = 0.0; }
    return;
  }
  let hc = cellH(ci, cj);
  let C = P.Cm * hc;
  let G = P.Gm * hc;
  let Un = vec4f(U(ci, cj), U(ci + 1, cj), U(ci + 1, cj + 1), U(ci, cj + 1));
  let Vn = vec4f(V(ci, cj), V(ci + 1, cj), V(ci + 1, cj + 1), V(ci, cj + 1));
  let W0n = vec4f(W0(ci, cj), W0(ci + 1, cj), W0(ci + 1, cj + 1), W0(ci, cj + 1));
  let Wn = vec4f(W(ci, cj), W(ci + 1, cj), W(ci + 1, cj + 1), W(ci, cj + 1)) + W0n; // total shape
  let q = 0.25 * P.hx * P.hy;
  let ihx = 1.0 / P.hx; let ihy = 1.0 / P.hy;
  let g3 = 0.57735026918962576;
  var Fu = vec4f(0.0); var Fv = vec4f(0.0); var Fw = vec4f(0.0);
  for (var gp = 0; gp < 4; gp++) {
    var xi = -g3; if ((gp & 1) != 0) { xi = g3; }
    var eta = -g3; if ((gp & 2) != 0) { eta = g3; }
    let dNdx = vec4f(-(1.0 - eta), 1.0 - eta, 1.0 + eta, -(1.0 + eta)) * (0.5 * ihx);
    let dNdy = vec4f(-(1.0 - xi), -(1.0 + xi), 1.0 + xi, 1.0 - xi) * (0.5 * ihy);
    let ux = dot(dNdx, Un); let uy = dot(dNdy, Un);
    let vx = dot(dNdx, Vn); let vy = dot(dNdy, Vn);
    let wx = dot(dNdx, Wn); let wy = dot(dNdy, Wn);
    let w0x = dot(dNdx, W0n); let w0y = dot(dNdy, W0n);
    // Marguerre strains: total slope minus the stress-free slope.
    let exx = ux + 0.5 * (wx * wx - w0x * w0x);
    let eyy = vy + 0.5 * (wy * wy - w0y * w0y);
    let gxy = uy + vx + wx * wy - w0x * w0y;
    let Nxx = C * (exx + P.nu * eyy);
    let Nyy = C * (eyy + P.nu * exx);
    let Nxy = G * gxy;
    Fu -= q * (Nxx * dNdx + Nxy * dNdy);
    Fv -= q * (Nyy * dNdy + Nxy * dNdx);
    Fw -= q * ((Nxx * wx + Nxy * wy) * dNdx + (Nxy * wx + Nyy * wy) * dNdy);
  }
  for (var a = 0u; a < 4u; a++) {
    cellF[base + a * 3u] = Fu[a];
    cellF[base + a * 3u + 1u] = Fv[a];
    cellF[base + a * 3u + 2u] = Fw[a];
  }
}

// True for the plate nodes of the drumstick contact patch (same rule as patchAround in params.ts).
fn inContactPatch(i: i32, j: i32) -> bool {
  if (nodeWeight(i, j) == 0.0) { return false; }
  let dxs = nodeX(i) - P.strikeX; let dys = nodeY(j) - P.strikeY;
  return dxs * dxs + dys * dys <= P.strikeR2 || id(i, j) == P.strikeNode;
}
// True for the plate nodes of the mount patch.
fn inMount(i: i32, j: i32) -> bool {
  if (P.mountOn == 0u || nodeWeight(i, j) == 0.0) { return false; }
  let dxm = nodeX(i) - P.mountX; let dym = nodeY(j) - P.mountY;
  return dxm * dxm + dym * dym <= P.mountR2 || id(i, j) == P.mountNode;
}

// Stored membrane force on local node `a` of cell (ci, cj).
fn cellForce(ci: i32, cj: i32, a: u32) -> vec3f {
  let base = cidx(ci, cj) * 12u + a * 3u;
  return vec3f(cellF[base], cellF[base + 1u], cellF[base + 2u]);
}

@compute @workgroup_size(16, 16)
fn main(@builtin(local_invocation_id) lid: vec3u, @builtin(workgroup_id) wid: vec3u) {
  let n = P.n;
  ox = i32(wid.x) * TILE - HALO;
  oy = i32(wid.y) * TILE - HALO;
  ihx2 = 1.0 / (P.hx * P.hx); ihy2 = 1.0 / (P.hy * P.hy); ihxy = 1.0 / (P.hx * P.hy);
  let lt = lid.y * 16u + lid.x;

  // Phase 1: one entry of the tile with halo per thread.
  {
    let gi = ox + i32(lid.x); let gj = oy + i32(lid.y);
    var ww = 0.0; var uu = 0.0; var vv = 0.0; var z0 = 0.0; var hh = 0.0;
    if (gi >= 0 && gi < i32(P.nx) && gj >= 0 && gj < i32(P.ny)) {
      let k = id(gi, gj);
      let st = nd[k];
      ww = posIn[2u * n + k]; uu = posIn[k]; vv = posIn[n + k]; z0 = st.x; hh = st.z;
    }
    tw[lt] = ww; tu[lt] = uu; tv[lt] = vv; tw0[lt] = z0; th[lt] = hh;
    if (lt < 64u) {
      var word = 0u;
      for (var q = 0u; q < 4u; q++) {
        let e = lt * 4u + q;
        let ei = ox + i32(e % u32(TW)); let ej = oy + i32(e / u32(TW));
        var wq = 0u;
        if (ei >= 0 && ei < i32(P.nx) && ej >= 0 && ej < i32(P.ny)) { wq = u32(round(4.0 * nd[id(ei, ej)].y)); }
        word |= wq << (q * 8u);
      }
      tn[lt] = word;
    }
  }
  workgroupBarrier();

  // Phase 2: membrane forces of the CT x CT cells, bending moments at the MT x MT
  // inner nodes, mixed differences at the cells; one item per thread each.
  if (lt < 169u) {
    let ci = ox + 1 + i32(lt % u32(CT)); let cj = oy + 1 + i32(lt / u32(CT));
    cellForces(ci, cj, lt);
    cc[lt] = Cc(ci, cj);
  }
  if (lt < 196u) {
    let mi = ox + 1 + i32(lt % u32(MT)); let mj = oy + 1 + i32(lt / u32(MT));
    ma[lt] = Ma(mi, mj);
    mb[lt] = Mb(mi, mj);
  }
  workgroupBarrier();

  // Phase 3: the update of the TILE x TILE nodes (no barriers past this point).
  if (lt >= 144u) { return; }
  let i = ox + HALO + i32(lt % u32(TILE)); let j = oy + HALO + i32(lt / u32(TILE));
  if (i >= i32(P.nx) || j >= i32(P.ny)) { return; }
  let k = id(i, j);
  let weight = NW(i, j);
  if (weight == 0.0) {
    // Outside the plate: the node stays at rest.
    posOut[k] = 0.0; posOut[n + k] = 0.0; posOut[2u * n + k] = 0.0; posOut[3u * n + k] = 0.0;
    velOut[k] = 0.0; velOut[n + k] = 0.0; velOut[2u * n + k] = 0.0;
    return;
  }

  // Elastic forces: bending from the discrete Kirchhoff energy, membrane from the cells.
  let bend = (ma[midx(i - 1, j)] - 2.0 * ma[midx(i, j)] + ma[midx(i + 1, j)]) * ihx2
           + (mb[midx(i, j - 1)] - 2.0 * mb[midx(i, j)] + mb[midx(i, j + 1)]) * ihy2
           + 2.0 * (1.0 - P.nu) * (cc[cidx(i, j)] - cc[cidx(i - 1, j)] - cc[cidx(i, j - 1)] + cc[cidx(i - 1, j - 1)]) * ihxy;
  var F = vec3f(0.0, 0.0, -P.hx * P.hy * bend);
  F += cellForce(i, j, 0u);
  F += cellForce(i - 1, j, 1u);
  F += cellForce(i - 1, j - 1, 2u);
  F += cellForce(i, j - 1, 3u);
  let pOld = vec3f(posIn[k], posIn[n + k], posIn[2u * n + k]);
  let vel = vec3f(velIn[k], velIn[n + k], velIn[2u * n + k]);
  // The mount: a spring and a dashpot to ground over the mount patch, on all three components.
  let mounted = inMount(i, j);
  if (mounted) { F -= P.mountKw * weight * pOld + P.mountCw * weight * vel; }
  let mass = P.rho * TH(i, j) * P.hx * P.hy * weight;
  var acc = F * (1.0 / mass);
  // Optional scaling of the in-plane inertia (lengthens the stable time step).
  acc.x *= P.invInPlaneMass; acc.y *= P.invInPlaneMass;

  // Losses: frequency-independent on all components, the graph Laplacian of the
  // transverse velocity (~k^2), and an extra frequency-independent in-plane loss.
  acc -= 2.0 * SP.sigma0 * vel;
  acc.z += 2.0 * SP.sigma1 * lapDW(i, j);
  acc.x -= 2.0 * SP.sigmaIn * vel.x;
  acc.y -= 2.0 * SP.sigmaIn * vel.y;

  // Drumstick: Hertz contact between a free mass and the contact patch (the
  // plate nodes within the tip radius, plus the nearest one). The contact
  // coordinate is the mass-weighted mean displacement of the patch, and the
  // force is spread over the patch in proportion to the node masses, so the
  // pair is conjugate and the contact has a potential energy.
  if ((SP.flags & 1u) != 0u) {
    let sIn = 4u * SP.parity;
    if (P.mountOn == 0u) {
      // Free plate: inertia relief. The force of the previous step is applied
      // to the whole plate as the rigid acceleration field that balances the
      // patch load and its moments about the center, so the plate gains no net
      // momentum (the thread would take it up in reality). The one-step lag
      // integrates to zero. A mounted plate takes the reaction through the mount.
      let Fprev = stick[sIn + 2u];
      acc.z += Fprev * (1.0 / P.mass + nodeY(j) * P.patchCY / P.Ix + nodeX(i) * P.patchCX / P.Iy);
    }
    if (inContactPatch(i, j)) {
      let sx = stick[sIn]; let sv = stick[sIn + 1u]; let sMax = stick[sIn + 3u];
      var wc = 0.0;
      for (var pj = i32(P.patchJ0); pj <= i32(P.patchJ1); pj++) {
        for (var pi = i32(P.patchI0); pi <= i32(P.patchI1); pi++) {
          if (inContactPatch(pi, pj)) { wc += nodeMass(pi, pj) * posIn[2u * n + id(pi, pj)]; }
        }
      }
      wc /= P.patchMass;
      let pen = wc - sx; // the stick sits on the +z side and moves toward -z
      var Fc = 0.0;
      if (pen > 0.0) { Fc = P.K * pow(pen, P.alpha); }
      acc.z -= Fc / P.patchMass;
      if (k == P.strikeNode) {
        let sOut = 4u * (1u - SP.parity);
        let svn = sv + P.dt * Fc / P.stickMass;
        stick[sOut] = sx + P.dt * svn;
        stick[sOut + 1u] = svn;
        stick[sOut + 2u] = Fc;
        stick[sOut + 3u] = max(sMax, Fc);
      }
    }
  }

  let vNew = vel + P.dt * acc;
  if (SP.audioSlot >= 0) { hist[u32(SP.audioSlot) * n + k] = vNew.z; }
  var pNew = pOld + P.dt * vNew;
  // Compensated update of w: the increment dt * v can be far below the
  // float32 resolution of w, so the rounding error is carried separately.
  let y = P.dt * vNew.z + posIn[3u * n + k];
  let t = pOld.z + y;
  let lo = (pOld.z - t) + y;
  pNew.z = t;
  posOut[k] = pNew.x; posOut[n + k] = pNew.y; posOut[2u * n + k] = pNew.z; posOut[3u * n + k] = lo;
  velOut[k] = vNew.x; velOut[n + k] = vNew.y; velOut[2u * n + k] = vNew.z;
}