{
 "schema_version": 1,
 "submission_id": "vonkarman-modal",
 "challenge_id": "cymbal-tapered-v1",
 "submitter": {
  "name": "Jeremy Magland",
  "email": "jmagland@flatironinstitute.org"
 },
 "date": "2026-09-17",
 "description": "Marguerre shallow-shell reduction of the challenge model on a Cartesian grid at 8 mm and 4 mm spacing, as for the hammered cymbal, with the thickness taper entering the mass, the bending stiffness and the membrane compliance 1/(E h) as Gauss-point coefficients of the assembled operators; the Airy modes are the eigenfunctions of the variable-coefficient compatibility operator. Full modal basis (the dents break the symmetry): 200 and 400 modes at 8 mm, 400 and 600 at 4 mm. The dents are a millimeter deep in a rim that is 0.7 mm thick, and the 8 mm cells barely resolve them, so the 8 mm members are coarse. The hole is closed (a stated approximation); the bell flank exceeds the shallow-shell range; the staircase boundary makes the radiated ring converge only at first order; and the response is chaotic beyond about 0.15 s, so errors between any two solutions have a floor of roughly 0.6 (waveform) and 0.5 (spectrogram).",
 "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-w200-a200-m2",
   "output": "vonkarman-modal/vk-g8-w200-a200-m2.wav",
   "method": "Marguerre shallow shell (Airy form) on a Cartesian grid masked to the disk, modal method on shell modes built from Bogner-Fox-Schmit plate modes with variable thickness, Gautschi-type integrator exact in the linear part, retarded Rayleigh-type integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.008,
    "cells": 2024,
    "n_dof": 8500,
    "transverse_modes": 200,
    "basis_even_in_y": false,
    "airy_modes": 200,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 10644.3
   },
   "timing_seconds": 72.53,
   "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-w400-a400-m2",
   "output": "vonkarman-modal/vk-g8-w400-a400-m2.wav",
   "method": "Marguerre shallow shell (Airy form) on a Cartesian grid masked to the disk, modal method on shell modes built from Bogner-Fox-Schmit plate modes with variable thickness, Gautschi-type integrator exact in the linear part, retarded Rayleigh-type integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.008,
    "cells": 2024,
    "n_dof": 8500,
    "transverse_modes": 400,
    "basis_even_in_y": false,
    "airy_modes": 400,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 16678.6
   },
   "timing_seconds": 399.021,
   "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-w400-a400-m2",
   "output": "vonkarman-modal/vk-g4-w400-a400-m2.wav",
   "method": "Marguerre shallow shell (Airy form) on a Cartesian grid masked to the disk, modal method on shell modes built from Bogner-Fox-Schmit plate modes with variable thickness, Gautschi-type integrator exact in the linear part, retarded Rayleigh-type integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.004,
    "cells": 8112,
    "n_dof": 33268,
    "transverse_modes": 400,
    "basis_even_in_y": false,
    "airy_modes": 400,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 12904.8
   },
   "timing_seconds": 478.696,
   "more_accurate_than": [
    "vk-g8-w400-a400-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-w600-a600-m2",
   "output": "vonkarman-modal/vk-g4-w600-a600-m2.wav",
   "method": "Marguerre shallow shell (Airy form) on a Cartesian grid masked to the disk, modal method on shell modes built from Bogner-Fox-Schmit plate modes with variable thickness, Gautschi-type integrator exact in the linear part, retarded Rayleigh-type integral with cubic fractional delays",
   "parameters": {
    "grid_spacing_m": 0.004,
    "cells": 8112,
    "n_dof": 33268,
    "transverse_modes": 600,
    "basis_even_in_y": false,
    "airy_modes": 600,
    "oversampling": 2,
    "dt_s": 1.0416666666666666e-05,
    "highest_transverse_mode_hz": 14584.7
   },
   "timing_seconds": 1364.02,
   "more_accurate_than": [
    "vk-g4-w400-a400-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-w600-a600-m2.h5",
    "format": "hdf5"
   }
  }
 ]
}