Perfectly Matched Layers for the Convected Helmholtz Equation

Abstract

Abstract. In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Bérenger's perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for timedependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented. 1. Introduction. Perfectly matched layers (PMLs) were introduced by Bérenger [3] in order to design efficient numerical absorbing boundary conditions (more precisely, absorbing layers) for the computation of time-dependent solutions of Maxwell's equations in unbounded domains. They have since been used for numerous applications, mostly in the time domain [4,28,5,23] but also for time-harmonic wave-like equations [27,15].In particular, PMLs have been used for the solution in the time domain of the linearized Euler equations [19,13,16,26], which model acoustic propagation in the presence of a flow. In this case, it has been observed that PMLs can lead to instabilities, due to the presence of waves whose phase and group velocities have opposite signs [26] (see [2] for a general analysis of this phenomenon). Some techniques have been developed to overcome this difficulty, making the layers stable but, unfortunately, no longer perfectly matched [16,1]. More recently, ideas for designing stable PMLs for this problem have emerged from several teams independently. These new approaches, which seem to be very closely related, have been developed for time-dependent applications in [20,11,14] and for time-harmonic applications in the present paper. These different works all deal with the case of a parallel flow, which is orthogonal to the layers.

New constructions of perfectly matched layers for the linearized Euler equations

“…The y) for t < T s and zero for t > T s with T s = 0.05, f c = 4/T s and δ M is the Dirac mass located in the middle of the computational domain. The right-hand side was f (t, x, y) on all three equations of system (1). For an oblique velocity u 0 = v 0 = 270, pressure near the upperleft corner is shown on Fig.…”

“…Among the two operators L and G, the only operator which generates waves propagating in both positive x and negative x directions is the operator L. This suggests that designing a PML for the Euler equation can be reduced to the design of a PML for the advective wave operator L. This question has been the subject of several works [1,3] and references therein. Following these works, we use for operator L a PML defined by replacing the x derivatives by a 'pml' x derivative.…”

Isogeometric analysis of sound propagation through laminar flow in 2-dimensional ducts

“…(32) for a given frequency. On the right-most truncation boundary γ + , we either use the mode matching formulation, or we insert a perfectly matched layer [15,36,33,37]. In the latter case, we attach a PML region of width 25% of the computational domain, let the absorption coefficient in this PML increase linearly with the spatial x-coordinate, and manually scale the absorption coefficient to minimize the artificial reflections.…”

“…The treatment of the artificial truncation boundaries γ ± is less trivial [33,15,16]. Here, waves must be allowed to propagate out of the domain, and, at the same time, reflections back into the domain must be avoided.…”

“…[12][13][14]. Finite element methods represent a popular numerical approach, based on, e.g., the convected Helmholtz equation [15,16], the linearized Euler equations [17], or the so-called Galbrun's equation [18][19][20]. These methods are often discretized using Lagrange elements with C 0 -continuity of the state variable approximations.…”