Note the staircase radiation problem in the submission and README
2 changed files+2−2
README.mdmodified+1−1View file
@@ -23,7 +23,7 @@ bash solvers/vonkarman-modal-disk/run_chain.sh <cymbal-repo>/challenges/cymbal-f
2323 bash solvers/vonkarman-modal-disk/run_chain.sh <cymbal-repo>/challenges/cymbal-flat-disk-v1/challenge.json gpu # 4 mm members, needs a CUDA GPU and PyTorch
2424 ```
2525
26-The chain is `g8-w100-a100-m1`, `g8-w200-a200-m1`, `g4-w200-a200-m2`, `g4-w300-a300-m2`, `g4-w450-a450-m2`, `g4-w600-a600-m4` (grid spacing in mm, transverse modes, Airy modes, time steps per output sample). Because the strike is centered on the x axis the solution is even in y, and the modal bases are restricted to that symmetric subspace, which is exact and halves the number of modes needed for a given frequency reach: 200 symmetric modes reach about 6 kHz, 300 about 9 kHz, 450 about 13.5 kHz and 600 about 18 kHz. The cost of a time step is the memory bandwidth of the cubic coupling tensor (8 bytes times the cube of the mode count, 1.7 GB at 600 modes), so the 8 mm members were run on a laptop CPU and the 4 mm members on a workstation GPU with `--device cuda`, where a step takes a few milliseconds at most; the two code paths agree to round-off, and the hardware is recorded per solution. The 8 mm grid has 2024 cells and 8500 degrees of freedom, the 4 mm grid 8112 cells and 33268. On the 8 mm grid the lowest elastic frequencies are 21.76 and 21.88 Hz for the two-nodal-diameter pair (21.87 Hz analytically), 38.01 Hz for the first axisymmetric mode (38.03) and 87.6 and 90.2 Hz for the four-nodal-diameter pair (89.61); on the 4 mm grid the last pair is 89.03 and 89.19 Hz. The staircase splits degenerate pairs by about a percent at 8 mm and a few tenths of a percent at 4 mm. The linear limit of the nonlinear solver agrees with the closed-form linear modal solution on the same mesh to 5e-4 at one step per sample and 1.3e-4 at two.
26+The chain is `g8-w100-a100-m1`, `g8-w200-a200-m1`, `g4-w200-a200-m2`, `g4-w300-a300-m2`, `g4-w450-a450-m2`, `g4-w600-a600-m4` (grid spacing in mm, transverse modes, Airy modes, time steps per output sample). Because the strike is centered on the x axis the solution is even in y, and the modal bases are restricted to that symmetric subspace, which is exact and halves the number of modes needed for a given frequency reach: 200 symmetric modes reach about 6 kHz, 300 about 9 kHz, 450 about 13.5 kHz and 600 about 18 kHz. The cost of a time step is the memory bandwidth of the cubic coupling tensor (8 bytes times the cube of the mode count, 1.7 GB at 600 modes), so the 8 mm members were run on a laptop CPU and the 4 mm members on a workstation GPU with `--device cuda`, where a step takes a few milliseconds at most; the two code paths agree to round-off, and the hardware is recorded per solution. The 8 mm grid has 2024 cells and 8500 degrees of freedom, the 4 mm grid 8112 cells and 33268. On the 8 mm grid the lowest elastic frequencies are 21.76 and 21.88 Hz for the two-nodal-diameter pair (21.87 Hz analytically), 38.01 Hz for the first axisymmetric mode (38.03) and 87.6 and 90.2 Hz for the four-nodal-diameter pair (89.61); on the 4 mm grid the last pair is 89.03 and 89.19 Hz. The staircase splits degenerate pairs by about a percent at 8 mm and a few tenths of a percent at 4 mm. The linear limit of the nonlinear solver agrees with the closed-form linear modal solution on the same mesh to 5e-4 at one step per sample and 1.3e-4 at two. A caution: the free disk's low modes radiate to the listener as a near-cancelling sum, and the staircase edge breaks the cancellation. In the linear closed-form solution the ring level after the strike falls from 0.040 Pa rms at 8 mm to 0.016 at 6 mm, 0.012 at 4 mm and 0.0096 at 3 mm, while the mode frequencies agree to within a percent, so the radiated pressure converges only at first order in the grid spacing and the 8 mm members ring several times too loud. This is the main reason the errors between the 8 mm and 4 mm members are large, and it is why a boundary-conforming (polar or mapped) grid is the natural next solver for this challenge.
2727
2828 ## Adding a submission
2929
solutions/cymbal-flat-disk-v1/vonkarman-modal.jsonmodified+1−1View file
@@ -7,7 +7,7 @@
77 "email": "jmagland@flatironinstitute.org"
88 },
99 "date": "2026-09-17",
10- "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.",
10+ "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 the strike falls from 0.040 Pa rms at 8 mm to 0.016 at 6 mm, 0.012 at 4 mm and 0.0096 at 3 mm, while the mode frequencies agree to within a percent. The 8 mm members therefore ring several times too loud and the 4 mm members are probably still 20 to 30 percent too loud; the large waveform and spectrogram errors between the two grids reflect this, not the number of modes. A boundary-conforming discretization is needed for the radiated pressure to converge.",
1111 "code_url": "https://vault1.magland.org/challenges/cymbal-solutions/blob/main/solvers/vonkarman-modal-disk/vk_modal_disk.py",
1212 "viz_url": "https://vault1.magland.org/challenges/cymbal-solutions/site/",
1313 "solutions": [