Advertisement

PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach

  • Jonas Ballani
  • Phuong Huynh
  • David Knezevic
  • Loi Nguyen
  • Anthony T. PateraEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We present an SCRBE PDE App framework for accurate and interactive calculation and visualization of the parametric dependence of the pressure field and associated Quantities of Interest (QoI)—such as impedance and transmission loss—for an extensive family of acoustic duct models. The Static Condensation Reduced Basis Element (SCRBE) partial differential equation (PDE) numerical approach incorporates several principal ingredients: component-to-system model construction, underlying “truth” finite element PDE discretization, (Petrov)-Galerkin projection, static condensation at the component level, parametrized model-order reduction for both the inter-component (port) and intra-component (bubble) degrees of freedom, and offline-online computational decompositions; we emphasize in this paper reduced port spaces and QoI evaluation techniques, especially frequency sweeps, particularly germane to the acoustics context. A PDE App constitutes a Web User Interface (WUI) implementation of the online, or deployed, stage of the SCRBE approximation for a particular parametrized model: User model parameter inputs to the WUI are interpreted by a PDE App Server which then invokes a parallel cloud-based SCRBE Online Computation Server for calculation of the pressure and associated QoI; the Online Computation Server then downloads the spatial field and scalar outputs (as a function of frequency) to the PDE App Server for interrogation and visualization in the WUI by the User. We present several examples of acoustic-duct PDE Apps: the exponential horn, the expansion chamber, and the toroidal bend; in each case we verify accuracy, demonstrate capabilities, and assess computational performance.

Notes

Acknowledgements

We thank Professor Masayuki Yano of University of Toronto for his contributions to the formulation and verification of the PDE Apps, Dr Sylvain Vallaghé of Akselos for his generous assistance in the FE verification of SCRBE resonances, Professor Kathrin Smetana of the University of Twente for valuable discussions related to the transfer eigenproblem, Professor Peter Dahl of University of Washington and Professor Jer-Ming Chen of Singapore University of Technology and Design (SUTD) for insightful reviews of earlier acoustic PDE Apps, Thomas Leurent of Akselos for his strong support of PDE Apps for education, and Thuc Nguyen of Akselos for his contributions to the web platform. This work was supported by the Swiss Confederations Innovation Promotion Agency (CTI) under Grant 17802.1 PFIW-IW (JB), ONR Contracts N00014-11-1-0713 and N00014-17-1-2077, OSD/AFOSR Grant FA9550-09-0613, SUTD International Design Center, and an MIT Ford Professorship (ATP).

References

  1. 1.
    B. Almroth, P. Stern, F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16(5), 525–528 (1978)CrossRefGoogle Scholar
  2. 2.
    P.R. Amestoy, A. Guermouche, J.Y. L’Excellent, S. Pralet, Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    I. Babuška, R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9, 373–406 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Balay, W.D. Gropp, L. Curfman McInnes, B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, ed. by E. Arge, A.M. Bruaset, H.P. Langtangen (Birkhäuser Press, Boston, 1997), pp. 163–202CrossRefGoogle Scholar
  5. 5.
    M. Barrault, Y. Maday, N. Nguyen, A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. CR Acad. Sci. Paris Ser. I 339, 667–672 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Binev, A. Cohen, R. Dahmen, G. Petrova, P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457–1472 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D.T. Blackstock, Fundamentals of Physical Acoustics, 1st edn. (Wiley, Hoboken, 2000)Google Scholar
  8. 8.
    R.J. Craig, M. Bampton, Coupling of substructures for dynamic analyses. AIAA J. 3(4), 678–685 (1968)zbMATHGoogle Scholar
  9. 9.
    J.L. Eftang, A.T. Patera, Port reduction in component-based static condensation for parametrized problems: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng. 96(5), 269–302 (2013)zbMATHGoogle Scholar
  10. 10.
    S. Félix, J.-P. Dalmont, C.J. Nederveen, Effects of bending portions of the air column on the acoustical resonances of a wind instrument. J. Acoust. Soc. Am. 131(5), 4164–4172 (2012)CrossRefGoogle Scholar
  11. 11.
    U. Hetmaniuk, R. Lehoucq, A special finite element method based on component mode synthesis. Math. Model. Numer. Anal. 44(3), 401–421 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    W.C. Hurty, On the dynamics of structural systems using component modes. AIAA Paper No. 64–487 (1964)Google Scholar
  13. 13.
    D.B.P. Huynh, D.J. Knezevic, J.W. Peterson, A.T. Patera, High-fidelity real-time simulation on deployed platforms. Comput. Fluids 43(1), 74–81 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Huynh, D.J. Knezevic, A.T. Patera, A static condensation reduced basis element method: approximation and a posteriori error estimation. Math. Model. Numer. Anal. 47(1), 213–251 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    P. Huynh, D.J. Knezevic, A.T. Patera, A static condensation reduced basis element method: complex problems. Comput. Methods Appl. Mech. Eng. 259, 197–216 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Part I: The h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995)MathSciNetzbMATHGoogle Scholar
  17. 17.
    B.S. Kirk, J.W. Peterson, R.M. Stogner, G.F. Carey, libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 23(3–4), 237–254 (2006)CrossRefGoogle Scholar
  18. 18.
    Y. Maday, E.M. Rønquist, The reduced basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26(1), 240–258 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    K.J. McMahon, A comparison of the transfer matrix method and the finite element method for the claculation of the transmission loss in a single expansion chamber muffler. Master’s thesis, RPI Hartford, December 2014Google Scholar
  20. 20.
    J.L. Munjal, Acoustics of Ducts and Mufflers, 2nd edn. (Wiley, Hoboken, 2014)Google Scholar
  21. 21.
    A.K. Noor, J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980)CrossRefGoogle Scholar
  22. 22.
    A Pinkus, N-Widths in Approximation Theory (Springer Science and Business Media, New York, 1985)CrossRefGoogle Scholar
  23. 23.
    J.T. Post, E.L. Hixson, A modeling and measurement study of acoustic horns. Ph.D. Thesis, University of Texas at Austin, May 1994CrossRefGoogle Scholar
  24. 24.
    G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Selamet, Z.L. Ji, Acoustic attenuation performance of circular expansion chambers with extended inlet/outlet. J. Sound Vib. 223(2), 197–212 (1999)CrossRefGoogle Scholar
  26. 26.
    A. Selamet, P.M. Radavich, The effect of length on the acoustic attenuation performance of concentric expansion chambers: an analytical, computational and experimental investigation. J. Sound Vib. 201(4), 407–426 (1997)CrossRefGoogle Scholar
  27. 27.
    K. Smetana, A.T. Patera, Optimal local approximation spaces for component-based static condensation procedures. SIAM J. Sci. Comput. 38(5), A3318–A3356 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    K. Veroy, C. Prud’homme, D.V. Rovas, A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. AIAA Paper No. 2003–3847 (2003), pp. 1–18Google Scholar
  29. 29.
    E.L. Wilson, The static condensation algorithm. Int. J. Numer. Methods Eng. 8(1), 198–203 (1974)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jonas Ballani
    • 1
  • Phuong Huynh
    • 2
  • David Knezevic
    • 2
  • Loi Nguyen
    • 2
  • Anthony T. Patera
    • 3
    Email author
  1. 1.Swiss Federal Institute of Technology in LausanneLausanneSwitzerland
  2. 2.Akselos S.A.LausanneSwitzerland
  3. 3.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations