Development of 3D PUFEM with linear tetrahedral elements for the simulation of acoustic waves in enclosed cavities

Abstract

This work is concerned with the numerical simulation of sound pressure field in three-dimensional cavities in which absorbing materials are present. Standard techniques such as the Finite Element Method are known to be extremely demanding computationally when the frequency increases and thus limited to low frequency applications. To alleviate these difficulties, an alternative formulation based on the Partition of Unity Finite Element Method is proposed. The method involves enriching the approximation finite element space by expanding the acoustic pressure in a set of plane waves propagating in various directions over the unit sphere. Particular attention is devoted to the fast and accurate computation of highly oscillatory integrals which is required by the method. Convergence studies show that these wave finite elements allow to capture accurately the wave field with a number of degrees of freedom that only grows quadratically with the frequency yielding drastic data reduction compared to classical FEM. Results of practical interest are shown for the case of a sound source placed in a reverberation room with absorbing materials. This archived file is not the final published version of "M. Yang , E. Perrey-Debain , B. Nennig , J.-D. Chazot, Development of 3D PUFEM with linear tetrahedra elements for the simulation of acoustic waves in enclosed cavities.

Partition of Unity Finite Element Method applied to exterior problems with Perfectly Matched Layers

“…In practical terms, the method allows a drastic reduction of degrees of freedom when compared with standard discretization techniques whilst preserving acceptable accuracy. A natural extension of this work is to develop the method further for the simulation of three-dimensional acoustic waves in the spirit of [28]. From results of Sections 3 and 4, it is clear that there is room for improvement of the method by considering more sophisticated PML, radial or conformal [29] and in order to deal more efficiently with small geometrical features, and in this context, recent publications [30,31] seem to offer interesting approaches.…”

“…As explained before, in presence of small geometrical features, the mesh has to be refined accordingly in order to capture correctly the geometry of the scatterer, this is illustrated in Figure 14e. In those regions performances of the method are not optimal because, as stated in [28], the PUFEM enrichment with plane waves corresponds to the high frequency representation of waves and this requires, in order to be effective, that each element should span over many wavelengths. A closer analysis of the PUFEM mesh, as shown in Figure 14e, reveals that many elements are smaller than the wavelength, here k % 11 cm.…”

Numerical modeling of the underwater acoustic impact of offshore stations

“…This enables using coarser meshes and, therefore, reducing the computational cost. In addition, the pollution error is mitigated, increasing the accuracy of the simulations Melenk (1995); Melenk and Babuska (1996); Babuska and Melenk (1997); Mayer and Mandel (1997); Laghrouche and Bettess (2000); Ortiz and Sanchez (2001); Laghrouche et al (2002Laghrouche et al ( , 2003; Bettess et al (2003); Sugimoto et al (2003); Perrey-Debain et al (2004); Ortiz (2004); Laghrouche et al (2005); ; Wang et al (2012); Yang et al (2018).…”

“…In this chapter, we extend the PUM formulation presented in Chapter 4 to 3D problems. Similar to Laghrouche et al (2003); Perrey-Debain et al (2004); Yang et al (2015); Mahmood et al (2017); Yang et al (2018), we paste at each node a set of plane waves. However, we use the set of wave directions proposed in Leopardi (2006).…”

“…Massimi et al (2008) extend the previous formulation to the analysis of layered media, considering evanescent modes. Recently, Yang et al (2015Yang et al ( , 2018 improved the pioneer formulation Perrey-Debain et al (2004) by implementing an exact integration algorithm based on the successive use of the Green's theorem. The examples considered a uniform spatial distribution of the wavenumber and tetrahedral elements.…”