{
 "challenge_id": "cymbal-bell-v1",
 "title": "Bowed bronze disk with a bell and a mounting hole, struck with a drum stick, geometrically nonlinear: radiated pressure",
 "generated": "2026-09-17T20:36:15+00:00",
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 "submissions": [
  {
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   "submission_id": "vonkarman-modal",
   "challenge_id": "cymbal-bell-v1",
   "submitter": {
    "name": "Jeremy Magland",
    "email": "jmagland@flatironinstitute.org"
   },
   "date": "2026-09-17",
   "description": "Marguerre shallow-shell reduction of the challenge model (Airy stress function with the initial shape, in-plane inertia neglected, damping proportional to the full linearized stiffness) on a Cartesian grid at 8 mm and 4 mm spacing, as for the bow, with the bell included in the initial shape. The mounting hole is not represented: the solver closes it, neglecting the mass and stiffness of a 12 mm disk at the apex. The bell flank has a slope of 0.46, beyond the comfort zone of shallow-shell theory, so these solutions carry a modeling error in the bell region in addition to the discretization error. Shell modes are built from the Bogner-Fox-Schmit plate modes of the staircase domain by diagonalizing the membrane stiffening once (the lowest mode is sensitive to the Airy count because of the bell's tight curvature); bases even in y; two-step integrator exact in the linear part; retarded integral of the normal acceleration with distances to the curved top face. Cautions as for the flat disk and the bow: first-order convergence of the radiated ring on staircase grids, and a chaotic response beyond about 0.15 s that puts a floor of roughly 0.6 (waveform) and 0.5 (spectrogram) under the errors between any two solutions.",
   "code_url": "https://vault1.magland.org/challenges/cymbal-solutions/blob/main/solvers/vonkarman-modal-disk/vk_modal_disk.py",
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   "key": "cymbal-solutions/vonkarman-modal",
   "source": "cymbal-solutions",
   "source_url": "https://vault1.magland.org/challenges/cymbal-solutions"
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 ]
}