Description

Context

Numerical tools that simulate wave propagation phenomena are widely used to address important industrial/societal challenges such as aircraft noise reduction, electromagnetic compatibility testing, and seismic risk assessment. When a limited number of frequencies need to be studied, frequency-domain methods based on the numerical solution of time-harmonic equations, of which the Helmholtz equation is the typical example, are generally considered. Unfortunately, it is extremely difficult to develop tools that are both fast and reliable for solving time-harmonic problems. It is even more complicated for the high-frequency cases, where the wavelength is small compared to the characteristic size of the problem.

The WavesDG project deals with the high-performance finite element solution of large-scale time-harmonic problems. Finite element methods (FEMs) can provide high-fidelity solutions to problems with complicated physical and geometric configurations. They are versatile and based on a strong mathematical foundation. However, discretizing time-harmonic problems with FEMs leads to large sparse linear systems that are difficult to solve with standard iterative schemes: They are ill-conditioned due to intrinsic properties of time-harmonic problems. The parallel iterative solution of these systems can be accelerated by using domain decomposition methods (DDMs), but standard DDM approaches are not efficient for time-harmonic problems and wave-specific strategies have to be developed.

Goals and resarch axes

In the WavesDG project, we will study and extend discontinuous Galerkin (DG) finite element schemes for the Helmholtz equation. These methods rely on discontinuous basis functions and transmission conditions (or penalty terms) prescribed at the interface between the elements to enforce weakly the continuity of the numerical solution. We will investigate DG methods with new wave-specific interface treatments, based on low-order and high-order transmission conditions. We will combine them with wave-specific DDMs and, in the last part of the project, with wave-specific basis functions. By using wave-specific strategies at all levels, we aim to improve the accuracy of the numerical solution and the properties of the linear system for computationally efficient solution schemes on distributed memory parallel architectures.

The research axes of the WavesDG project address the systematic numerical study of standard/non-standard DG methods (including a discussion of the transmission operators and the coupling variables) and the mathematical study of the most promising methods in order to strengthen the mathematical foundation of the methods and to guide the parameters selection. Standard high-order polynomial basis functions will be considered in the first parts of this project, and plane-wave basis functions will be investigated in the last part to provide comprehensive wave-specific solution procedures. Selected wave-specific DG schemes will be implemented in a dedicated parallel 3D code, accelerated with a wave-specific DDM. This code will be used to evaluate the efficiency of the approach on realistic large-scale time-harmonic acoustic problems.

Some references from the team

  • Boubendir, Antoine, Geuzaine (2012). A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation. JCP 231 (2), 262-280
  • Bériot, Prinn, Gabard (2016). Efficient implementation of high‐order finite elements for Helmholtz problems. IJNME 106(3), 213-240
  • Bériot, Modave (2021). An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape. IJNME, 122(5), 1239-1261
  • Chaumont-Frelet, Nicaise (2020). Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems. IMA JNA, 40(2), 1503-1543
  • Chaumont-Frelet, Valentin (2020). A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains. SIAM JNA, 58(2), 1029-1067
  • Gabard (2007). Discontinuous Galerkin methods with plane waves for time-harmonic problems. JCP, 225(2):1961–1984
  • Lieu, Gabard, Bériot (2016). A comparison of high-order polynomial and wave-based methods for Helmholtz problems. JCP, 321, 105-125
  • Lieu, Marchner, Gabard, Bériot, Antoine, Geuzaine (2020). A non-overlapping Schwarz domain decomposition method with high-order finite elements for flow acoustics. CMAME, 369, 113223
  • Modave, Geuzaine, Antoine (2020). Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering. JCP, 401, 109029
  • Modave, Royer, Antoine, Geuzaine (2020). A non-overlapping domain decomposition method with high-order transmission cond. and cross-point treatment for Helmholtz pbms. CMAME, 368, 113162
  • Modave, St-Cyr, Mulder, Warburton (2015). A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters. GJI, 203 (2), 1419-1435.
  • Prinn (2014). Efficient finite element methods for aircraft engine noise prediction. Doctoral dissertation, University of Southampton