Publication on crack identification in membranes based on eigenfrequencies

9 Oktober 2026

Our new article, “Can one hear the shape of a crack in a drum? Spectral fingerprints and a data-driven perspective”, by Thomas Apel, Serge Nicaise, and Philipp Zilk has been published in the SeMA Journal, the official journal of the Spanish Society of Applied Mathematics (SeMA).

The study investigates whether the geometric properties of a crack in a membrane can be determined from the corresponding eigenfrequencies. Using the short-time asymptotic expansion of the heat trace, we first prove analytically that both the existence and the length of a crack can be inferred from the Laplace spectrum of the domain. Under additional assumptions, further parameters, such as the angle of a crack, can also be reconstructed.

In addition, we present a data-driven approach: a neural network is trained on highly accurate spectral data obtained from numerical simulations to determine the corresponding crack parameters from eigenfrequencies. The simulations are based on isogeometric analysis, with the singularities at the crack tips resolved using graded mesh refinement and error estimates ensuring the accuracy of the computed spectra. Evaluation on a test set shows that the trained network can reliably reconstruct the crack parameters.

The publication thus presents a novel approach to crack identification that combines analytical results, accurate numerical simulations, and machine-learning methods.

Apel, T., Nicaise, S. & Zilk, P. Can one hear the shape of a crack in a drum? Spectral fingerprints and a data-driven perspective. SeMA (2026). https://doi.org/10.1007/s40324-026-00444-9