Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations

2026-08-06

AI Summary

This study addresses the challenge of efficiently and stably simulating nonlinear Föppl–von Kármán plate vibrations, which is hindered by the high computational cost of modal coupling. The authors propose an explicit energy-stable method based on modal synthesis: nonlinear terms are evaluated in physical space using a pseudospectral approach, while derivatives are handled exactly in the modal domain. Discrete sine and cosine transforms naturally enforce simply supported boundary conditions, and a scalar auxiliary variable enables explicit time integration. This approach is the first within a modal framework to simultaneously achieve explicit stability, spectral accuracy in frequency representation, and computational efficiency, substantially reducing the overhead associated with nonlinear modal coupling. The method’s efficacy is demonstrated through high-fidelity acoustic simulations.

Abstract

Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

Problem

• Föppl-von Kármán equations
• modal synthesis
• nonlinear coupling
• computational cost
• pseudospectral method

Innovation

• pseudospectral method
• modal synthesis
• Föppl-von Kármán equations
• scalar auxiliary variable
• nonlinear plate dynamics