Structure preserving algorithms for simulation of linearly damped acoustic systems

Vasileios Chatziioannou1

1Department of Music Acoustics, University of Music and Performing Arts Vienna, Austria
Received: 22 August 2017 ; Accepted in revised form: 22 January 2020

Abstract: Energy methods for constructing time-stepping algorithms are of increased in- terest 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.

c 2019 European Society of Computational Methods in Sciences and Engineering
Keywords: Energy conservation, symplectic form, mechanical integrator, linear damping
Mathematics Subject Classification: 65M06, 65P10, 65Z05


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