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.
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Notes
- 1.
A first realization of the PDE App concept is provided in [13], however the latter invokes a much simpler and less general model order reduction approach—the Reduced Basis method but without components—that is furthermore restricted by an “onboard” implementation which is much less powerful than the web cloud formulation described in the current paper.
- 2.
We note that Eq. (2) (IW), and also Eq. (2) (R) in the Sommerfeld case, can be slightly improved: for Eq. (2) (IW) in the term ik(p − 2p inc) and for Eq. (2) (R) in the term ikp we might replace k with \(k^{\mathrm {d}} = k/\sqrt {1 + \text{i} k \epsilon }\). However, the dissipation term is extremely small, and hence the effect of this improvement correspondingly unimportant.
- 3.
The Minimal Component Dataset is not small, but it is loaded by the WUI only upon launch of a particular parametrized model; the per-query response is thus not affected.
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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).
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Ballani, J., Huynh, P., Knezevic, D., Nguyen, L., Patera, A.T. (2019). PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_1
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