{
 "challenge_id": "cymbal-flat-disk-v1",
 "title": "Flat bronze disk struck with a drum stick, geometrically nonlinear: radiated pressure",
 "generated": "2026-09-17T20:36:16+00:00",
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   "submission_id": "vonkarman-modal",
   "challenge_id": "cymbal-flat-disk-v1",
   "submitter": {
    "name": "Jeremy Magland",
    "email": "jmagland@flatironinstitute.org"
   },
   "date": "2026-09-17",
   "description": "Von Karman thin-plate reduction (Airy stress function, in-plane inertia neglected, damping on bending only) of the challenge model on a Cartesian grid: the disk is approximated by the square cells whose centers lie inside it (a staircase boundary) at 8 mm and 4 mm spacing. Linear free-plate modes and clamped Airy modes come from Bogner-Fox-Schmit finite elements on the cells and a sparse shift-invert eigensolver; the bases are restricted to functions even in y, which is exact because the strike is centered on the x axis. The cubic coupling tensor is integrated on the cells, the time stepping is a two-step scheme exact in the linear part, and the retarded Rayleigh integral uses cubic fractional delays. The chain is ordered by grid spacing, number of modes, and time step; the mode truncation caps the frequency the cascade can reach. Caution on the staircase: the free disk's low modes radiate as a near-cancelling sum, and the staircase edge breaks that cancellation. In the linear closed-form solution on the same meshes the ring level after a strike falls from 8 mm to 6, 4 and 3 mm roughly as 4 : 1.7 : 1.25 : 1, while the mode frequencies agree to within a percent, so the 8 mm members ring several times too loud and the 4 mm members are probably still 20 to 30 percent too loud; the large errors between the two grids reflect this, not the number of modes. A boundary-conforming discretization is needed for the radiated pressure to converge.",
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