Structure preserving algorithms for simulation of linearly damped acoustic systems

TL;DR

This paper extends energy-preserving and symplectic algorithms to non-conservative acoustic systems with linear damping, deriving discrete conservation laws and analyzing symplectic structure evolution, demonstrated through a vibrating string case study.

Contribution

It introduces novel structure-preserving algorithms for damped acoustic systems, expanding energy and symplectic methods beyond conservative Hamiltonian frameworks.

Findings

Derived discrete energy conservation laws for damped systems

Analyzed symplectic structure evolution in discrete and continuous cases

Validated methods with a vibrating string bouncing against an obstacle

Abstract

Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic integrators may be constructed that preserve the symplectic form of the system. This methodology has been established for Hamiltonian systems, with numerous applications in engineering problems. In this paper an extension of such methods to non-conservative acoustic systems is presented. Discrete conservation laws, equivalent to that of energy-conserving schemes, are derived for systems with linear damping, incorporating the action of external forces. Furthermore the evolution of the symplectic structure is analysed in the continuous and the discrete case. Existing methods are examined and novel methods are designed using a lumped oscillator as an elemental model. The proposed methodology is extended to the case of distributed systems and exemplified through a case study of a vibrating string bouncing against a rigid obstacle.