Simulation of the Geometrically Exact Nonlinear String via Energy Quadratisation

TL;DR

This paper introduces a linearly-implicit numerical scheme for simulating nonlinear string vibrations that accurately captures large stretchings and wave interactions while conserving energy, improving computational efficiency over existing methods.

Contribution

It proposes a novel energy quadratisation-based scheme that efficiently handles nonlinear effects and separates wave directions, enhancing simulation accuracy and stability.

Findings

Conserves a quadratic energy sum during simulation

Accurately resolves transverse and longitudinal wave speeds

Demonstrates improved computational efficiency in numerical experiments

Abstract

String vibration represents an active field of research in acoustics. Small-amplitude vibration is often assumed, leading to simplified physical models that can be simulated efficiently. However, the inclusion of nonlinear phenomena due to larger string stretchings is necessary to capture important features, and efficient numerical algorithms are currently lacking in this context. Of the available techniques, many lead to schemes which may only be solved iteratively, resulting in high computational cost, and the additional concerns of existence and uniqueness of solutions. Slow and fast waves are present concurrently in the transverse and longitudinal directions of motion, adding further complications concerning numerical dispersion. This work presents a linearly-implicit scheme for the simulation of the geometrically exact nonlinear string model. The scheme conserves a numerical energy, expressed as the sum of quadratic terms only, and including an auxiliary state variable yielding the nonlinear effects. This scheme allows to treat the transverse and longitudinal waves separately, using a mixed finite difference/modal scheme for the two directions of motion, thus allowing to accurately resolve the wave speeds at reference sample rates. Numerical experiments are presented throughout.