{
 "schema_version": 1,
 "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.",
 "code_url": "https://vault1.magland.org/challenges/cymbal-solutions/blob/main/solvers/vonkarman-modal-disk/vk_modal_disk.py",
 "viz_url": "https://vault1.magland.org/challenges/cymbal-solutions/site/",
 "solutions": [
  {
   "solution_id": "vk-g8-w100-a100-m2",
   "output": "vonkarman-modal/vk-g8-w100-a100-m2.wav",
   "method": "von Karman plate (Airy form) on a Cartesian grid masked to the disk, modal method on Bogner-Fox-Schmit modes, Gautschi-type integrator exact in the linear part, retarded Rayleigh integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.008,
    "cells": 2024,
    "n_dof": 8500,
    "transverse_modes": 100,
    "basis_even_in_y": true,
    "airy_modes": 100,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 2776.3
   },
   "timing_seconds": 56.155,
   "more_accurate_than": [],
   "hardware": "Intel(R) Xeon(R) w5-3435X (32 threads), numpy/scipy for the setup; coupling tensor and time stepping on NVIDIA RTX 6000 Ada Generation (PyTorch 2.14.0+cu130)"
  },
  {
   "solution_id": "vk-g8-w200-a200-m2",
   "output": "vonkarman-modal/vk-g8-w200-a200-m2.wav",
   "method": "von Karman plate (Airy form) on a Cartesian grid masked to the disk, modal method on Bogner-Fox-Schmit modes, Gautschi-type integrator exact in the linear part, retarded Rayleigh integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.008,
    "cells": 2024,
    "n_dof": 8500,
    "transverse_modes": 200,
    "basis_even_in_y": true,
    "airy_modes": 200,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 5832.6
   },
   "timing_seconds": 48.943,
   "more_accurate_than": [
    "vk-g8-w100-a100-m2"
   ],
   "hardware": "Intel(R) Xeon(R) w5-3435X (32 threads), numpy/scipy for the setup; coupling tensor and time stepping on NVIDIA RTX 6000 Ada Generation (PyTorch 2.14.0+cu130)"
  },
  {
   "solution_id": "vk-g4-w200-a200-m2",
   "output": "vonkarman-modal/vk-g4-w200-a200-m2.wav",
   "method": "von Karman plate (Airy form) on a Cartesian grid masked to the disk, modal method on Bogner-Fox-Schmit modes, Gautschi-type integrator exact in the linear part, retarded Rayleigh integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.004,
    "cells": 8112,
    "n_dof": 33268,
    "transverse_modes": 200,
    "basis_even_in_y": true,
    "airy_modes": 200,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 5890.6
   },
   "timing_seconds": 64.792,
   "more_accurate_than": [
    "vk-g8-w200-a200-m2"
   ],
   "hardware": "Intel(R) Xeon(R) w5-3435X (32 threads), numpy/scipy for the setup; coupling tensor and time stepping on NVIDIA RTX 6000 Ada Generation (PyTorch 2.14.0+cu130)"
  },
  {
   "solution_id": "vk-g4-w300-a300-m2",
   "output": "vonkarman-modal/vk-g4-w300-a300-m2.wav",
   "method": "von Karman plate (Airy form) on a Cartesian grid masked to the disk, modal method on Bogner-Fox-Schmit modes, Gautschi-type integrator exact in the linear part, retarded Rayleigh integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.004,
    "cells": 8112,
    "n_dof": 33268,
    "transverse_modes": 300,
    "basis_even_in_y": true,
    "airy_modes": 300,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 9000.3
   },
   "timing_seconds": 203.828,
   "more_accurate_than": [
    "vk-g4-w200-a200-m2"
   ],
   "hardware": "Intel(R) Xeon(R) w5-3435X (32 threads), numpy/scipy for the setup; coupling tensor and time stepping on NVIDIA RTX 6000 Ada Generation (PyTorch 2.14.0+cu130)",
   "fields": {
    "path": "vonkarman-modal/vk-g4-w300-a300-m2.h5",
    "format": "hdf5"
   }
  }
 ]
}