Hold the shell's six rigid motions: the shallow-shell strains are not rotation invariant and a free, tilting shell is unphysical
4 changed files+6−6
MODEL.mdmodified+3−3View file
@@ -65,7 +65,7 @@ $$
6565 \rho h\, (w_{tt} + 2 \sigma_0 w_t) = \partial_\alpha \left[ N_{\alpha\beta} (Z_\beta + w_\beta) \right] - \partial_{\alpha\beta} M_{\alpha\beta} - \sum_i F_i(t)\, \phi_i,
6666 $$
6767
68-together with the free-edge conditions that the weak form implies at $r = R$ and $r = r_h$. The shell is at rest at $t = 0$: $u = v = w = 0$ and $u_t = v_t = w_t = 0$.
68+together with the free-edge conditions that the weak form implies at $r = R$ and $r = r_h$. The rigid motions of the shell are suppressed. Let $\langle \mathbf{a}, \mathbf{b} \rangle = \int_A \rho h\, \mathbf{a} \cdot \mathbf{b}\, dA$ be the mass-weighted inner product of displacement fields $(u, v, w)$, and let $\mathcal{R}$ be the span of the six rigid fields $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, $(0, 0, x)$, $(0, 0, y)$ and $(-y, x, 0)$. The solution is orthogonal to $\mathcal{R}$ at all times, $\langle (u, v, w), \mathbf{r} \rangle = 0$ for every $\mathbf{r} \in \mathcal{R}$, and the weak form above is required only for test functions orthogonal to $\mathcal{R}$. This is the shell held by an ideal support that fixes its position and orientation and constrains nothing else: the reaction is a body force in $\mathcal{R}$, equal to $\rho h$ times the rigid acceleration field that balances the net force and moments of the strikes, and the elastic motion is untouched. The shell is at rest at $t = 0$: $u = v = w = 0$ and $u_t = v_t = w_t = 0$.
6969
7070 ## The strikes
7171
@@ -105,7 +105,7 @@ with $w_{tt}$ taken as zero at negative times. This is the field of a planar sou
105105
106106 ## Remarks
107107
108-The shell is free. The strains are invariant under the small rigid motions $w \to w + a + \theta_\alpha x_\alpha$, $u_\alpha \to u_\alpha - \theta_\alpha (Z + w) - \tfrac{1}{2} \theta_\alpha \theta_\beta x_\beta$ (with $u_x = u$, $u_y = v$), so the momentum and angular momentum that the strikes impart appear as a drift of $w$ that is linear in $t$ between contacts, has no transverse acceleration, and does not enter $p$; the in-plane inertia of that drift is of relative order $\rho \Omega^2 R^2 / E$, about $10^{-8}$. The transverse displacement itself does contain the drift.
108+The rigid constraint is not a convenience. The shallow-shell strains are not invariant under finite rotations: a shell that has tilted by an angle $\theta$ behaves in this model like a shell of slope $\theta$, with a membrane-bending coupling that a real rotated plate does not have. Left free, the shell tilts by about a radian within half a second of the first strike, its partials glide and its treble drains into in-plane motion, none of it physical. Holding the six rigid components at zero removes the problem exactly, and the strikes' momentum and angular momentum go into the support.
109109
110110 In the linear regime about the flat state, $\sigma_0$ damps every mode at the rate $\sigma_0$ and $\eta$ damps a mode of angular frequency $\omega$ at the rate $\eta \omega^2 / 2$.
111111
@@ -155,4 +155,4 @@ The reference is chosen automatically among the submitted solutions by the rule
155155
156156 ## Optional field export
157157
158-A submission may provide the transverse displacement for visualization; it is not scored. The file is HDF5 with datasets `x` and `y`, float64, shape (61,), $x_i = -R + 2R\, i / 60$ and the same for $y$; `t`, float64, shape (3501,), $t_k = k / 500$; and `u`, float32, shape (3501, 61, 61), with `u[k, j, i]` $= w(x_i, y_j, t_k)$ in meters for grid points in $A$ and NaN elsewhere. The drift of the free shell is part of $w$. Root attributes `challenge_id`, `solution_id`, `quantity` and `units` are recommended. The file is about 50 MB and belongs in git LFS or at an external URL.
158+A submission may provide the transverse displacement for visualization; it is not scored. The file is HDF5 with datasets `x` and `y`, float64, shape (61,), $x_i = -R + 2R\, i / 60$ and the same for $y$; `t`, float64, shape (3501,), $t_k = k / 500$; and `u`, float32, shape (3501, 61, 61), with `u[k, j, i]` $= w(x_i, y_j, t_k)$ in meters for grid points in $A$ and NaN elsewhere. Root attributes `challenge_id`, `solution_id`, `quantity` and `units` are recommended. The file is about 50 MB and belongs in git LFS or at an external URL.
README.mdmodified+1−1View file
@@ -2,7 +2,7 @@
22
33 This repository defines two simulation challenges on one geometrically nonlinear thin-shell model of a 16 inch crash cymbal struck by a drum stick, and publishes the comparison of every registered solution at [challenges/cymbal2/site](https://vault1.magland.org/challenges/cymbal2/site/). It follows the arrangement of the [string](https://vault1.magland.org/challenges/string), [plate](https://vault1.magland.org/challenges/plate) and [cymbal](https://vault1.magland.org/challenges/cymbal) repositories: the model is defined in the continuum, solutions live in separate registered repositories, and the shared [challenge-tools](https://vault1.magland.org/challenges/challenge-tools) package scores them against a reference chosen from among the submissions.
44
5-The model is defined once, in [MODEL.md](MODEL.md): a shallow shell with in-plane inertia, Saint Venant-Kirchhoff through the thickness, Kelvin-Voigt and mass-proportional damping, free at both edges, struck by a point mass through a Hertz contact, radiating to a listener 1.5 m away through the Rayleigh integral. Each challenge fixes the profile and the thickness. In both, the same stick lands three times, two seconds apart: on the bell (or where the bell would be), at half the radius, and near the rim, so that the difference in sound between a ping on the bell and a crash near the rim comes out of the same equations within one seven-second recording.
5+The model is defined once, in [MODEL.md](MODEL.md): a shallow shell with in-plane inertia, Saint Venant-Kirchhoff through the thickness, Kelvin-Voigt and mass-proportional damping, free at both edges with its six rigid motions held, struck by a point mass through a Hertz contact, radiating to a listener 1.5 m away through the Rayleigh integral. Each challenge fixes the profile and the thickness. In both, the same stick lands three times, two seconds apart: on the bell (or where the bell would be), at half the radius, and near the rim, so that the difference in sound between a ping on the bell and a crash near the rim comes out of the same equations within one seven-second recording.
66
77 | ID | Geometry | Deliverable |
88 | --- | --- | --- |
challenges/cymbal2-crash-v1/challenge.jsonmodified+1−1View file
@@ -72,7 +72,7 @@
7272 },
7373 "fields": {
7474 "format": "hdf5",
75- "quantity": "transverse displacement w(x, y, t) of the mid-surface, including the rigid drift; NaN outside the annulus",
75+ "quantity": "transverse displacement w(x, y, t) of the mid-surface; NaN outside the annulus",
7676 "units": "m",
7777 "axes": [
7878 "t",
challenges/cymbal2-flat-v1/challenge.jsonmodified+1−1View file
@@ -68,7 +68,7 @@
6868 },
6969 "fields": {
7070 "format": "hdf5",
71- "quantity": "transverse displacement w(x, y, t) of the mid-surface, including the rigid drift; NaN outside the annulus",
71+ "quantity": "transverse displacement w(x, y, t) of the mid-surface; NaN outside the annulus",
7272 "units": "m",
7373 "axes": [
7474 "t",