function result = plate_fd(params)
% PLATE_FD  Explicit finite differences for a Kirchhoff-Love plate in its mount, under gravity.
%
%   result = plate_fd(params) advances the damped plate equation
%
%     mu w_tt + 2 mu sigma0 w_t - 2 mu sigma1 lap(w_t) + D lap(lap(w))
%         - div(N grad w) = F(t) s(x,y) + mu g_n + support forces
%
%   on a uniform nx x ny grid with a second-order explicit scheme, the spatial
%   operators being the sparse energy-method matrices of plate_bending (the
%   13-point biharmonic stencil inside, the free-edge conditions as natural
%   conditions on the edges) and plate_prestress (the in-plane stress of the
%   plate's own weight).  The mount (plate_mount) supplies the support forces:
%   a thread is a transverse spring at its hole, a pad or a felt a one-sided
%   spring and dashpot, a vise holds a band of nodes still, fingers are a
%   two-sided spring over a patch.  The plate starts from its static
%   equilibrium (plate_static) and the departure from it is carried as
%   rigid-body coordinates plus a flexural field (plate_rigid_basis), so that
%   the pendulum swing of a hanging plate and its flexural vibration come out
%   of one time loop.
%
%   The step is dt = cfl * dt_max with dt_max from plate_model; the scheme is
%   unstable for cfl > 1 and the run stops as soon as the field diverges,
%   reporting the step at which it did (info.diverged_step).
%
%   The work is in three stages, each its own file, so that a host can run the
%   time loop elsewhere (the lab has a WebGPU port of plate_fd_loop) while the
%   setup and the output stay this code:
%     S      = plate_fd_setup(params)    grid, step, statics, load, probe, listener, operators
%     Rec    = plate_fd_loop(S)          the time stepping; records probe, rigid coordinates,
%                                        support forces, frames, audio bins
%     result = plate_fd_output(S, Rec)   stresses, audio, spectra, the result struct
%
%   Approximations.  Second-order truncation error in dx, dy, dt; free edges
%   and curved edges first order at the boundary; small-angle theory for the
%   threads and the pendulum motion; the probe is snapped to the nearest grid
%   node; stress fields are evaluated only at the saved frames; the audio is
%   an estimate of the waveform, not a calibrated level.
%
%   See plate_result_schema for the fields of result.

S = plate_fd_setup(params);
Rec = plate_fd_loop(S);
result = plate_fd_output(S, Rec);
end