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Technical Brief

Numerical Local Time Stepping Solutions for Transient Statistical Energy Analysis

[+] Author and Article Information
Oriol Guasch

Associate Professor
GTM—Grup de Recerca en Tecnologies Mèdia,
Department of Engineering,
La Salle, Universitat Ramon Llull,
C/Quatre Camins 2,
Barcelona, Catalonia 08022, Spain
e-mail: oguasch@salleurl.edu

Carlos García

INNOVATION—Test and Validation,
Alstom Wind S.L.U.,
Barcelona, Catalonia 08005, Spain
e-mail: carlos.garcia-martinez@power.alstom.com

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 2, 2014; final manuscript received August 17, 2014; published online September 19, 2014. Assoc. Editor: Lonny Thompson.

J. Vib. Acoust 136(6), 064502 (Sep 19, 2014) (7 pages) Paper No: VIB-14-1036; doi: 10.1115/1.4028454 History: Received February 02, 2014; Revised August 17, 2014

Subsystem energies evolve in transient statistical energy analysis (TSEA) according to a linear system of ordinary differential equations (ODEs), which is usually numerically solved by means of the forward Euler finite difference scheme. Stability requirements pose limits on the maximum time step size to be used. However, it has been recently pointed out that one should also consider a minimum time step limit, if time independent loss factors are to be assumed. This limit is based on the subsystem internal time scales, which rely on their characteristic mean free paths and group velocities. In some cases, these maximum and minimum limits become incompatible, leading to a blow up of the forward Euler solution. It is proposed to partially mitigate this problem by resorting to a local time-stepping finite difference strategy. Subsystems are grouped into sets characterized by different time step sizes and evolve according to them.

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Figures

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Fig. 1

Local time-stepping algorithm strategies

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Fig. 3

Energy time evolution for the left cavity in Fig. 2. — local time-stepping, … exact solution, - - forward Euler finite differences. (a) 250 Hz, (b) 500 Hz, (c) 1000 Hz, (d) 2000 Hz, (e) 4000 Hz, and (f) 8000 Hz.

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Fig. 5

tstab versus τd/2 for the numerical example. –×tstab, ×τd/2 for the cavity, - -τd/2 for concrete. Gray area indicates admissible values for the time step △t according to Eq. (7).

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Fig. 4

Energy time evolution for the wall X221 in Fig. 2. local time-stepping, … exact solution, - - forward Euler finite differences. (a) 250 Hz, (b) 500 Hz, (c) 1000 Hz, (d) 2000 Hz, (e) 4000 Hz, and (f) 8000 Hz.

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