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Stabilized Time-Discontinuous Galerkin Methods With Applications to Structural Acoustics

[+] Author Affiliations
Lonny L. Thompson

Clemson University

Prapot Kunthong

Kasetsart University

Paper No. IMECE2006-15753, pp. 31-40; 10 pages
doi:10.1115/IMECE2006-15753
From:
  • ASME 2006 International Mechanical Engineering Congress and Exposition
  • Noise Control and Acoustics
  • Chicago, Illinois, USA, November 5 – 10, 2006
  • Conference Sponsors: Noise Control and Acoustics Division
  • ISBN: 0-7918-4776-4 | eISBN: 0-7918-3790-4
  • Copyright © 2006 by ASME

abstract

The time-discontinuous Galerkin (TDG) method possesses high-order accuracy and desirable C-and L-stability for second-order hyperbolic systems including structural acoustics. C- and L-stability provide asymptotic annihilation of high frequency response due to spurious resolution of small scales. These non-physical responses are due to limitations in spatial discretization level for large-complex systems. In order to retain the high-order accuracy of the parent TDG method for high temporal approximation orders within an efficient multi-pass iterative solution algorithm which maintains stability, generalized gradients of residuals of the equations of motion expressed in state-space form are added to the TDG variational formulation. The resultant algorithm is shown to belong to a family of Pade approximations for the exponential solution to the spatially discrete hyperbolic equation system. The final form of the algorithm uses only a few iteration passes to reach the order of accuracy of the parent solution. Analysis of the multi-pass algorithm shows that the first iteration pass belongs to the family of (p+1)-stage stiff accurate Singly-Diagonal-Implicit-Runge-Kutta (SDIRK) method. The methods developed can be viewed as a generalization to the SDIRK method, retaining the desirable features of efficiency and stability, now extended to high-order accuracy. An example of a transient solution to the scalar wave equation demonstrates the efficiency and accuracy of the multi-pass algorithms over standard second-order accurate single-step/single-solve (SS/SS) methods.

Copyright © 2006 by ASME

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