Modal sound synthesis from surface meshes, built on a meshfree thin-shell
solver. A triangle mesh is interpreted as a thin shell (drum-shell, cymbal,
plate) of a chosen material and thickness; chladni assembles its
Kirchhoff–Love stiffness with one of three discretizations — Loop
subdivision FEM, first-order local max-entropy (LME), or second-order
max-entropy (SME) — solves the generalized eigenproblem for the elastic
modes, projects a hammer impulse onto them, and renders the result as realtime
audio and a slow-motion mesh deformation with a Chladni nodal-line overlay.
mesh.obj + IsotropicMaterial (E, nu, rho) + thickness h + loss factor eta
│
▼
thin-shell stiffness assembly ── one of three formulations, behind a
│ single ShellAssembler interface:
│
├─ LME / SME (default) meshfree curved Kirchhoff–Love. Local
│ max-entropy basis on a k-ring chart per node (Arroyo–Ortiz 2006);
│ curved-shell bending strain (Millán–Rosolen–Arroyo 2011 §3.3);
│ 2nd-order graded-nodal-gap variant on by default
│ (Rosolen–Millán–Arroyo 2013 §3.2.2). Ghost nodes on free edges.
├─ Loop Loop-subdivision FEM on the C¹ limit surface
│ (Cirak–Ortiz–Schröder 2000), exact extraordinary-vertex evaluation
│ (Stam 1999). Rejects meshes with boundary valence ∉ {2,3,4}.
└─ legacy CST + IBM constant-strain membrane + isometric bending
(Grinspun 2003 / Wardetzky 2007); fallback only when Loop rejects.
│
▼
mass matrix M consistent (or lumped) vertex masses → 3n × 3n sparse
│
▼
generalized eigenproblem K φ = ω² M φ (Spectra shift-invert,
│ multi-seed + Rayleigh–Ritz fusion to span
│ degenerate clusters; rigid-body modes filtered)
▼
impulse projection a_k = (φ_k · f / ω_k) · H_hammer · H_band
│
▼
resonator-bank synthesis z_k(0) = (a_k, 0); z(t+dt) = z(t)·exp((-d+iω)dt)
│ sample = Σ_k Im(z_k); decoupled audio thread
├──► audio (WAV via chladni::wav, realtime via miniaudio)
└──► visual (Polyscope: extruded shell, slow-motion deformation,
signed-normal amplitude overlay with isolines / nodal lines)
State is value-typed: material, mesh, and per-formulation parameters are plain
structs, and the eigensolve produces a self-contained mode bank with no
globals. The three formulations live behind ShellAssembler
(include/chladni/shell/assembler.hpp): the LME/SME basis and curved
Kirchhoff–Love assembly are in src/shell/lme.cpp, Loop subdivision in
src/shell/loop.cpp (with exact limit-surface evaluation in
src/shell/loop_stam.cpp), and dispatch plus the legacy CST+IBM fallback in
src/shell/assembler.cpp.
cmake -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build -j
ctest --test-dir build --output-on-failureThe first configure fetches Eigen, Spectra, Catch2, tinyobjloader, and (if
CHLADNI_BUILD_GUI=ON, the default) Polyscope + miniaudio from source — no
system packages required. Polyscope's dependency tree (GLFW, glad, Dear ImGui,
glm, json) makes the first build take a few minutes; incremental builds touch
only the changed file. To build only the library + CLI demo:
cmake -B build -DCHLADNI_BUILD_GUI=OFFAPI documentation (Doxygen, with @cite resolved against the bibliography):
cmake -B build -DCHLADNI_BUILD_DOCS=ON
cmake --build build --target docs
open build/docs/html/index.htmlInteractive GUI (Polyscope viewer + ImGui controls + miniaudio output) — click the mesh to strike it; the resonator bank rings the modes out through the audio device while the mesh deforms in slow motion:
./build/apps/strike_gui/strike_gui # default mesh: models/cylinder.obj
./build/apps/strike_gui/strike_gui my.obj # arbitrary OBJ (or drag-and-drop)A formulation TabBar (Loop / LME / Legacy) selects the assembler at the top of the control panel; the default is the LME tab with 2nd-order SME enabled. The Single-mode excitation panel drives the textbook Chladni demo: pick a mode index, press Ring mode k, and with damping near zero the nodal isolines freeze into that mode's stationary Chladni figure.
CLI demo — loads a steel drum-shell cylinder, solves 30 free-free modes, and writes a 2.5 s mono WAV:
./build/apps/synth_demo/synth_demo /tmp/drum.wav
afplay /tmp/drum.wav # macOS — Linux: aplay / paplayHeadless tests (Catch2; every algorithm gated by an analytic or tabulated reference):
./build/tests/chladni_tests # 369 default cases
./build/tests/chladni_tests "[.slow]" # the gated convergence sweepsEach discretization is checked against closed-form or tabulated shell results, not just against the others. The summary below is the at-a-glance verdict; the per-resolution numbers behind it are in Method comparison.
| Benchmark | Reference | Measured result |
|---|---|---|
| Free-edge disk, 7 Leissa modes (3 topologies) | Leissa 1969, NASA SP-160 | LME 0.32–0.77 % max err vs shipped Loop 3.5–7.0 % — meshfree ~8–11× more accurate at matched DOF on bending-dominated open shells; barely splits degenerate doublets the mesh cannot resolve |
| Closed icosphere, n=2 spheroidal pentet | Wilkinson 1965 / Duffey 2005 cubic | All three converge O(h²) through k=5 (Loop 0.83 → 0.03 %, LME 6.78 → 0.092 %, SME 3.94 → 0.067 %). Loop wins on smooth closed shells — lowest error at every level, and cheaper |
| Free–free cylinder, n=2 ovalling | Rayleigh–Love inextensional | The off-label regime — the source papers only test diaphragm ends. Loop flat +4.2–4.5 %; SME +6.1 %, rising to +18.7 % only at the most axially-coarse aspect (classical membrane locking, HF-confirmed); first-order LME locks, with spurious low modes inflating the lowest-mode average past +100 % |
| Scordelis–Lo roof | shell-obstacle-course, w/w_ref = 0.3024 | SME → 1.002 at 1089 V — converges to unity on this membrane-dominated benchmark |
| Geodesic pinched hemisphere, 18° hole | Millán 2011 Fig 17, δ/δ_ref = 0.0924 | Loop cannot run (boundary valence 5/6 unsupported); SME flat ~1.10 from the coarsest level — the mesh-independence counterexample |
Per-resolution results behind the summary, across the three discretizations — Loop subdivision FEM, first-order LME, and second-order SME (the default). Bold marks the best method in each row.
Free-edge disk — Leissa, 7 modes through (1,1), max relative error.
| topology (V) | Loop (shipped) | LME | SME |
|---|---|---|---|
| polar 32×8 (257) | 6.52 % | 0.77 % | 1.00 % |
| iso n=32 (117) | 7.03 % | 0.62 % | 2.38 % |
| hex n=8 (211) | 3.54 % | 0.32 % | 3.63 % |
Loop (shipped) is the compute_shell_modes_loop default — 7-pt Dunavant
quadrature with consistent mass; tuning across 6 quadrature×lumping configs
lowers it to a best-of-6 of 4.82 / 6.44 / 3.54 %, still well above LME. The
(2,0) doublet split on the symmetry-free iso disk is LME 0.010 %, SME
0.027 %, Loop 0.39–0.58 % — meshfree barely splits a degenerate doublet the mesh
cannot resolve.
Closed icosphere — Wilkinson n=2 spheroidal pentet, relative error.
| k (V) | Loop (shipped) | LME | SME |
|---|---|---|---|
| 2 (162) | 0.83 % | 6.78 % | 3.94 % |
| 3 (642) | 0.27 % | 1.47 % | 0.96 % |
| 4 (2562) | 0.10 % | 0.36 % | 0.26 % |
| 5 (10242) | 0.03 % | 0.092 % | 0.067 % |
All three converge O(h²); meshfree starts 4–8× behind Loop at k=2 and closes steadily — to 2.6× by k=4 and ~2× by k=5 (SME 0.067 % vs Loop 0.03 %) — but never overtakes. Loop wins on smooth closed shells at every level. (The meshfree k=5 row was initially unobtainable: the solve was OOM-killed at 30,726 DOF, which looked like wide-stencil factorization fill but was in fact an assembly over-emission bug — the curved assemblers emitted ~30–70× more triplets than nonzeros; fixed, k=5 now runs in ~3 GB. See the Weaknesses note.) "Shipped" Loop here uses the multi-pass extraordinary-vertex path; the exact-evaluation Stam variant is 1.3–2.7× worse at every level.
Free–free cylinder — Rayleigh–Love n=2 ovalling, 24×n_along, relative error.
| aspect (V) | Loop | LME | SME |
|---|---|---|---|
| 2.39 (792) | +4.2 % | +172 % | +18.7 % |
| 1.59 (1176) | +4.4 % | +330 % | +8.8 % |
| 1.19 (1560) | +4.5 % | +116 % | +6.5 % |
| 1.01 (1848) | +4.5 % | +49 % | +6.1 % |
The off-label regime (the papers test diaphragm ends). The LME column is genuine membrane locking inflated by spurious low modes contaminating the lowest-mode average; SME's +18.7 % worst cell is the most axially-coarse aspect, easing to +6.1 % as the grid isotropizes. Loop's C¹ basis sidesteps the locking — see Weaknesses.
Scordelis–Lo roof — w / w_ref (reference 0.3024), diaphragm ends.
| V | Loop | LME | SME |
|---|---|---|---|
| 289 | 0.982 | 1.147 | 1.051 |
| 625 | 0.983 | 1.071 | 1.012 |
| 1089 | 0.984 | 1.039 | 1.002 |
Best = closest to the 1.0 target. SME converges to unity on this membrane-dominated benchmark, overtaking Loop (which sits ~1.6 % low) by 625 V.
Pinched hemisphere, 18° hole — δ / δ_ref (reference 0.0924), geodesic mesh.
| V | Loop | LME (γ=0.8) | SME (α=2) |
|---|---|---|---|
| 142 | cannot run | 1.428 | 1.104 |
| 301 | cannot run | 1.301 | 1.121 |
| 516 | cannot run | 1.244 | 1.115 |
| 1125 | cannot run | 1.198 | 1.104 |
| 1969 | cannot run | 1.178 | 1.101 |
This is the paper's own Fig-17 discretization. Loop rejects every level — its boundary stencil admits only valence-2/3/4 rim vertices, and the geodesic hole rim carries valence-5/6 ones; the meshfree schemes are valence-agnostic. SME is flat from the coarsest level (the Fig-17 character). On a degenerate lat-long mesh (kept as a mesh-quality contrast) the same fixture runs Loop 0.607 → 1.073, LME 2.297 → 1.325, SME 0.351 → 1.122 over 81 → 1089 V.
- Free-edge fidelity. On open bending-dominated shells the meshfree basis is sub-1 % against Leissa (LME 0.32–0.77 % across three disk topologies), versus 3.5–7.0 % for shipped Loop FEM at matched DOF — roughly 8–11× more accurate.
- Independent of mesh symmetry and connectivity. Meshfree barely splits degenerate doublets a procedural mesh cannot resolve (iso-disk (2,0) split: LME 0.010 % vs Loop 0.39–0.58 %), and it runs the paper's own geodesic pinched hemisphere — where Loop's boundary stencil rejects the valence-5/6 rim vertices outright. This is connectivity independence, not triangle-quality independence (see Weaknesses).
- Faithfulness-audited. Every numerical device that departs from the source papers is inventoried and justified, and the one invented feasibility net is dormant on all quality/clean meshes — its Newton-drop channel reads zero across the entire verification suite, so no benchmark cell rests on a non-paper mechanism.
- Three formulations, one interface. Loop / LME / SME (plus a legacy
fallback) share a single
ShellAssembler, so the same mesh and material can be re-solved under any discretization for direct comparison. The core library is graphics-free, and a Doxygen build with bibliography-resolved@citetags is wired into CMake.
- Sensitive to mesh quality (not symmetry). On a low-quality triangulation — e.g. a consistent-diagonal quad mesh with valence-4/8 vertices and ~15° slivers — meshfree modal fields are 2–4× rougher than Loop, because Loop subdivides (low-pass-filters the control mesh) while the meshfree basis is evaluated on the actual vertices. The fix is remeshing, not a code change; "mesh independence" here means symmetry/connectivity, not element quality.
- Membrane locking on thin curved shells — the classical shell-FEM regime. On the free–free cylinder (the one setup the source papers never test — they use diaphragm ends), low-order elements lock in the inextensional limit, exactly the difficulty that MITC / reduced-integration shell elements exist to address. SME stays at +6.1 %, rising to +18.7 % only at the most axially-coarse aspect (genuine, Hellmann–Feynman-confirmed, not removable by chart choice); first-order LME locks harder, with spurious low modes inflating the lowest-mode average past +100 %. Loop's C¹ subdivision basis sidesteps it (flat +4.2–4.5 %) — this is the regime where, of the three, Loop is the right tool.
- Cost profile. A per-Gauss-point Newton solve on a k-ring chart is orders of magnitude slower than Loop's local subdivision rules — that assembly cost is intrinsic to the meshfree basis. (A former wall here, an OOM-kill of the meshfree eigensolve at icosphere k=5, turned out to be a bug, not the factorization: the curved assemblers emitted one block per node-pair per triangle in their shared support — a ~30–70× over-emission, ~196 M triplets for a 2.9 M-nonzero matrix — which spiked peak memory past 18 GB. Folding per-thread before emit cut k=4 from 12 GB to 0.5 GB and unblocked k=5 at ~3 GB.) The shift-invert factorization does become the binding cost around k=6, where a factorization-free eigensolver (LOBPCG / Jacobi–Davidson) would be the next step.
- Legacy / unimplemented paths. The CST + IBM path is retained only as a fallback for meshes Loop rejects. A solid (tet / 3D elasticity) mode is not implemented — genuinely absent, not abandoned.
Meshfree max-entropy basis (the default formulation):
- Arroyo, M. and Ortiz, M. Local Maximum-Entropy Approximation Schemes: A Seamless Bridge Between Finite Elements and Meshfree Methods. Int. J. Numer. Methods Eng. 65(13):2167–2202, 2006. doi:10.1002/nme.1534. — the first-order LME basis.
- Millán, D., Rosolen, A., and Arroyo, M. Thin Shell Analysis from Scattered Points with Maximum-Entropy Approximants. Int. J. Numer. Methods Eng. 85(6):723–751, 2011. doi:10.1002/nme.2992. — the curved Kirchhoff–Love bending-strain formulation on point sets.
- Rosolen, A., Millán, D., and Arroyo, M. Second-Order Convex Maximum Entropy Approximants with Applications to High-Order PDE. Int. J. Numer. Methods Eng. 94(2):150–182, 2013. doi:10.1002/nme.4443. — the second-order (SME) approximants and the graded nodal-gap feasibility recipe.
Subdivision-surface FEM:
- Cirak, F., Ortiz, M., and Schröder, P. Subdivision Surfaces: A New Paradigm for Thin-Shell Finite-Element Analysis. Int. J. Numer. Methods Eng. 47(12):2039–2072, 2000. doi:10.1002/(SICI)1097-0207(20000430)47:12. — the Loop-subdivision shell elements.
- Stam, J. Evaluation of Loop Subdivision Surfaces. SIGGRAPH '99 Course Notes, 1999. — exact evaluation at extraordinary vertices.
Discrete-shells / legacy path:
- Grinspun, E., Hirani, A. N., Desbrun, M., and Schröder, P. Discrete Shells. SCA 2003, pp. 62–67. — the membrane + dihedral-bending energy.
- Wardetzky, M., Bergou, M., Harmon, D., Zorin, D., and Grinspun, E. Discrete Quadratic Curvature Energies. Computer Aided Geometric Design 24(8–9): 499–518, 2007. doi:10.1016/j.cagd.2007.07.006. — the isometric bending model.
- Tamstorf, R. and Grinspun, E. Discrete Bending Forces and Their Jacobians. Graphical Models 75(6):362–370, 2013. doi:10.1016/j.gmod.2013.07.001. — the closed-form analytic bending Hessian.
Analytic references for the verification benchmarks:
- Leissa, A. W. Vibration of Plates. NASA SP-160, 1969. — circular / annular free-edge plate frequencies.
- Leissa, A. W. Vibration of Shells. NASA SP-288, 1973. — cylindrical and spherical shell frequencies.
- Wilkinson, J. P. Natural Frequencies of Closed Spherical Shells. J. Acoust. Soc. Am. 38(2):367–368, 1965. doi:10.1121/1.1909710.
- Duffey, T. A., Pepin, J. E., Robertson, A. N., and Steinzig, M. L. Vibrations of Complete Spherical Shells with Imperfections. LA-UR-04-6904, IMAC-XXIII, 2005. — restates the Wilkinson cubic in legible form.
- Vogel, S. M. and Skinner, D. W. Natural Frequencies of Transversely Vibrating Uniform Annular Plates. J. Appl. Mech. 32(4):926–931, 1965. doi:10.1115/1.3627337.
- Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions. NBS Applied Mathematics Series 55, 1964. — Bessel-function approximations.
Modal sound synthesis:
- van den Doel, K. and Pai, D. K. The Sounds of Physical Shapes. Presence 7(4):382–395, 1998. — the resonator-bank synthesis primitive.
- O'Brien, J. F., Shen, C., and Gatchalian, C. M. Synthesizing Sounds from Rigid-Body Simulations. SCA 2002, pp. 175–181. — the impulse → modal amplitudes → audio pipeline.
- Bonneel, N., Drettakis, G., Tsingos, N., Viaud-Delmon, I., and James, D. Fast Modal Sounds with Scalable Frequency-Domain Synthesis. ACM TOG (Proc. SIGGRAPH 2008) 27(3):24:1–24:9, 2008.
Tools and libraries (fetched at configure time):
- Guennebaud, G., Jacob, B., et al. Eigen v3. eigen.tuxfamily.org — dense and sparse linear algebra.
- Qiu, Y., et al. Spectra. spectralib.org — sparse generalized eigensolver.
- Sharp, N., et al. Polyscope. polyscope.run — the GUI viewer.
- miniaudio — realtime audio output. tinyobjloader — OBJ mesh loading. Catch2 — the test framework.
include/chladni/ public headers (Doxygen-commented)
src/shell/ the three thin-shell assemblers (lme, loop, assembler)
src/ synth, mass, eigensolve, mesh, wav
tests/analytical/ plate / shell closed-form tests
tests/shell/ FEM building blocks + the method-comparison fixtures
tests/unit/ OBJ loader, synth, WAV writer
apps/synth_demo/ CLI: mesh → WAV
apps/strike_gui/ interactive: click-to-strike (Polyscope + miniaudio)
apps/chladni_figures/ headless: render the 6×6 nodal-pattern contact sheet
docs/papers/ references.bib — bibliography for the Doxygen @cite tags
images/ the rendered hero figure
models/ test meshes (cylinder.obj, torus.obj)
Free for personal, academic, and research use. No warranty. No commercial use.