Philippe Marchner
email: philippe.marchner@siemens.com

Davide Bizzarri
email: davide.bizzarri@siemens.com

Hadrien Bériot
email: hadrien.beriot@siemens.com

Improved low-frequency PML absorbing functions for exterior three dimensional Helmholtz problems

Keywords: Perfectly matched layer, low frequencies, Lambert function, absorbing functions, finite elements, exterior Helmholtz problem

The perfectly matched layer technique is routinely used and relied upon to solve time-harmonic radiation and scattering problems. Its purpose is to truncate the computational domain and absorb outgoing waves in order to mimic the Sommerfeld radiation condition. Thanks to its substantial development, the method has been successfully applied to many time-harmonic unbounded problems. While the perfectly matched layer is very effective in the high frequency regime, its accuracy can deteriorate at low frequencies, hampering the robustness of the method. This article investigates the low frequency limitation of the perfectly matched layer for three-dimensional Helmholtz exterior problems. To handle the issue we propose a novel family of parameter-free absorbing functions, that can be directly integrated in classical discretization schemes with minimal modifications. We assess our method on academic and industrial benchmarks in a finite element context, and show how the approach reduces the computational cost and modelling time in an engineering workflow.

Introduction

The accurate truncation of unbounded domains for wave propagation is a longstanding problem, with initial investigations going back to the late 1970s [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Bayliss | Radiation boundary conditions for wave-like equations[END_REF]. Problems posed in an unbounded domain must satisfy a radiation condition posed at infinity, which translates into a non-local boundary condition when the computation domain is truncated. These class of boundary conditions fall into the family of non-reflecting boundary conditions. A large variety of robust methods have been devised over the years. We refer for example to [START_REF] Tsynkov | Numerical solution of problems on unbounded domains. A review[END_REF][START_REF] Givoli | High-order local non-reflecting boundary conditions: a review[END_REF] for a review of historical developments. The design of non-reflecting boundary conditions is still an active research area and a design issue in simulation software, even for basic wave propagation models.

Among the non-reflecting boundary techniques, the Perfectly Matched Layer (PML) introduced in [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] has been widespread because of its versatility. It is a local method that extends the computational domain by a small volume layer and damps outgoing waves through a complex stretching applied to the spatial coordinates. The complex coordinate stretching was generalized in [START_REF] Collino | The perfectly matched layer in curvilinear coordinates[END_REF][START_REF] Teixeira | Complex space approach to perfectly matched layers: a review and some new developments[END_REF] and triggered many initiatives aimed at employing PMLs in domains of complex geometry. In that context, we can highlight the locally-conformal PML [START_REF] Ozgun | Non-Maxwellian locally-conformal PML absorbers for finite element mesh truncation[END_REF][START_REF] Ozgun | Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation[END_REF] and the Automatically Matched Layer [START_REF] Bériot | An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape[END_REF] approaches, that enable a standardized implementation suitable for industrial simulations. Although the perfectly matched layer has found applications extending beyond its initial purpose e.g. [START_REF] Zheng | A perfectly matched layer approach to the nonlinear schrödinger wave equations[END_REF][START_REF] Antoine | Towards perfectly matched layers for time-dependent space fractional PDEs[END_REF], it still faces many challenges for example in periodic [START_REF] Oskooi | The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers[END_REF] and anisotropic [START_REF] Bécache | Stability of perfectly matched layers, group velocities and anisotropic waves[END_REF] materials.

In addition, once the PML is discretized, it loses its reflectionless property at the continuous level and causes numerical reflections [START_REF] Johnson | Notes on Perfectly Matched Layers (PMLs)[END_REF][START_REF] Harari | Analytical and numerical studies of a finite element PML for the Helmholtz equation[END_REF][START_REF] Givoli | Computational absorbing boundaries[END_REF]. This is why the damping is typically applied gradually, and several efforts to optimize the PML parameters have been undertaken [START_REF] Modave | Optimizing perfectly matched layers in discrete contexts[END_REF][START_REF] Collino | Optimizing the perfectly matched layer[END_REF]. A notable advance in this direction was the usage of a singular, unbounded damping profile introduced by Bermúdez et al. [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF], enabling to recover the exact wavefield in the truncated domain at the continuous level. Using finite difference schemes, nearly-optimal PMLs based on rational interpolation were constructed in [START_REF] Druskin | Near-optimal perfectly matched layers for indefinite Helmholtz problems[END_REF], and a first layer design that cancels all numerical reflections has been achieved in [START_REF] Chern | A reflectionless discrete perfectly matched layer[END_REF] using discrete complex analysis.

This article focuses on improving the PML absorbing profile for three-dimensional exterior Helmholtz problems, with a focus on low frequencies. While it is well-known that the method converges exponentially with the layer width [START_REF] Lassas | Analysis of the PML equations in general convex geometry[END_REF], it is difficult to determine which accuracy can be achieved in a given numerical configuration. It is generally advised to position the PML at a minimum distance of half a wavelength from the obstacle [START_REF] Thompson | A review of finite-element methods for time-harmonic acoustics[END_REF], and to select its size sufficiently large with respect to the excitation frequency [START_REF] Bériot | On the locally-conformal perfectly matched layer implementation for Helmholtz equation[END_REF][START_REF] Cimpeanu | A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation[END_REF]. We will observe that these recommendations impose a severe limitation on the PML's ability to achieve a reasonable accuracy at low frequencies. This lack of clear guideline reveals a non-trivial behaviour of the wavefield in the PML, and can be overcome by using adaptive refinement techniques [START_REF] Chen | An adaptive perfectly matched layer technique for time-harmonic scattering problems[END_REF][START_REF] Michler | Improving the performance of perfectly matched layers by means of hp-adaptivity, Numerical Methods for Partial Differential Equations[END_REF].

In this article, we rather propose to revisit the unbounded function from Bermudez et al. [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF] by extending it to spherical waves, such as to approach a linear decay of the wavefield inside the PML for all frequencies. By doing so, we minimize discrete reflections when using a conventional finite element approach. We propose a simple, yet efficient methodology to improve the selection of the absorbing function, based on a systematic analysis of the interpolation errors in radial coordinates.

Our approach leads to approximations of the special Lambert function, which gives rise to damping profiles with both unbounded real and imaginary parts. As a result, evanescent waves are also damped and may accelerate the convergence of the PML. The approach can be related to non-linear transformations [START_REF] Hugonin | Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization[END_REF] or an infinite mapping layer [START_REF] Schoder | Revisiting infinite mapping layer for open domain problems[END_REF]. In this direction, the perfectly absorbing layer (PAL) method advocates the idea of a complex compression coordinate transformation for both evanescent and propagative modes [START_REF] Yang | A truly exact perfect absorbing layer for time-harmonic acoustic wave scattering problems[END_REF], albeit a rescaling of the unknown variable. Closely tied to our research, let us must mention the PhD work [START_REF] Deakin | Optimal PML transformations for the Helmholtz equation[END_REF] regarding optimal PML transformations. Finally when the nature of the wavefield is known at infinity, one may completely bypass domain truncation techniques by transforming the Helmholtz problem into another PDE [START_REF] Zenginoglu | A null infinity layer for wave scattering[END_REF][START_REF] Nabizadeh | Kelvin transformations for simulations on infinite domains[END_REF].

It is worth mentioning that the damping of low frequency waves have been investigated in detail in the time-domain, in particular for geophysics applications. Kuzuoglu and Mittra [START_REF] Kuzuoglu | Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers[END_REF] early proposed a solution, which has been improved through the complex frequency-shifted PML [START_REF] Roden | Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media[END_REF] and double-pole PML [START_REF] Correia | Performance of regular PML, CFS-PML, and second-order PML for waveguide problems[END_REF]. Such PMLs imply more parameters to be tuned, which is often done by heuristic considerations. We refer to the review [START_REF] Pled | Review and recent developments on the perfectly matched layer (PML) method for the numerical modeling and simulation of elastic wave propagation in unbounded domains[END_REF] for historical developments.

The article is organized as follows: we first highlight in Section 2 the low frequency limitation that arises at the discrete level when using common PML formulations for exterior Helmholtz problems. With the goal of targeting a linear decay of the wavefield for all frequencies, we derive and propose in Section 3 improved absorbing functions in either the low or high frequency regimes. In Section 4, we assess the performance of the new absorbing functions on simple academic problems in a finite element discretization context. We show that we can partially overcome the low frequency limitation with parameter-free functions, while maintaining the usual accuracy for high frequencies. We set up a variety of models in an industrial context in Section 5, where we show the potential of the approach to handle large frequency sweeps accurately at a low computational cost, with a single finite element mesh. We conclude by discussing perspectives and limitations of the method.

Low frequency limitation of the PML

Exterior problem setting

We consider a three dimensional Helmholtz radiation problem set in the unbounded domain Ω + = R 3 \Ω -, where Ω -is an open bounded set modeling a closed obstacle that we assume of regular boundary. Through all the article we stick to the +ıωt time-harmonic convention. We want to compute the acoustic wavefield u(x), x ∈ Ω + , solution of the boundary value problem

∆u + k 2 u = 0 in Ω + , ∂u ∂n = g on ∂Ω -, g ∈ L 2 (∂Ω -), lim r→∞ r ∂ ∂r + ık u = 0, r = ∥x∥,
where n is the outward unit normal to ∂Ω -, g is a given source function, and r ≥ 0 is the radial coordinate.

We have introduced the wavenumber k = 2πf /c 0 , with c 0 the speed of sound of the medium at rest, and f the running frequency. It is well-established that the outgoing solution to the radiation problem can be decomposed into spherical harmonics Y m ℓ [START_REF] Wilcox | A generalization of theorems of Rellich and Atkinson[END_REF],

u(r, θ, φ) = ∞ ℓ=0 ℓ m=-ℓ C mℓ h (2) ℓ (kr)Y m ℓ (θ, φ), (1)
where (r, θ, φ) are the spherical coordinates, h

0 (z) = ı e -ız z , h (2)
1 (z) = -(z -ı) e -ız z 2 , h (2)
2 (z) = -ı(z 2 -3ız -3) e -ız z 3 .
In the far field, the fundamental solution h

(2) 0 (z) is a spherical wave that satisfies the Sommerfeld radiation condition. It behaves locally as a plane wave. In the near field, at low frequencies, the series (1) typically requires only a few terms to converge, while at higher frequencies more terms may be necessary. In practice, we would like to solve the Helmholtz radiation problem in a bounded domain. To do so, we must introduce a domain truncation technique that reproduces the behaviour of the solution at infinity. In this article, we study the perfectly matched layer method, and focus on its numerical accuracy and computational cost for three-dimensional exterior Helmholtz problems.

The perfectly matched layer transformation for spherical waves

The perfectly matched layer method introduces a fictitious volume layer surrounding the truncated, bounded domain. The goal of the method is to make the solution decay in the layer without introducing back-reflections. Let us consider a spherical domain of radius r 0 . The PML appends a spherical layer defined in the range r ∈ [r 0 , r 0 +δ], where δ > 0 defines the width of the layer. The outgoing solution u is transformed in the layer by a complex change of variables of the form

r = r + F (r) ık , F (r) = r r0 σ(s)ds, r ∈]r 0 , r 0 + δ[, (2)
where σ(r) ≥ 0 is the absorbing function which is strictly increasing. For a plane wave at normal incidence u p (r) = e -ıkr , the transformation gives r) , such that the real part of F is responsible for the decay of the wave in the layer. The value F (r 0 + δ) should be sufficiently large, such that u p (r 0 + δ) is close to zero, and no back-reflections occur in the physical domain. Moreover, the perfect matching continuity relation u p (r 0 ) = u p (r 0 ) has to be satisfied, such that no reflections occur at the transition between the physical and PML domains. If these conditions are fulfilled, the Sommerfeld radiation condition is emulated at the continuous level. For a spherical wave of the form u s (r) = e -ıkr /r, a difficulty arises because the transformation also acts on the radial decay of the field. Applying transformation (2) gives

u p (r) = u p (r)e -F (
u s (r) = u s (r)d(r), d(r) = e -F (r) 1 + 1 ıkr F (r) , (3)
where we refer to the function d as the decay function. In this case, the choice of an appropriate absorbing function F is more difficult since the behaviour of the decay function is also driven by its denominator, in particular as kr → 0. This impacts the PML performance at the discrete level, an issue that we propose to address in this study. When kr is large, lim kr→+∞ d(r) = e -F (r) and we recover the plane wave behaviour. We show in Figures 1a and1b the transformation of the spherical wave u s (r) by two common PML mappings, at respectively wavenumbers of k = 64π and k = π/64. For the illustration, we choose the commonly used hyperbolic [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF] and cubic stretchings [START_REF] Collino | Optimizing the perfectly matched layer[END_REF], denoted respectively σ u and σ c

σ u (r) = 1 (r 0 + δ) -r , σ c (r) = σ r -r 0 δ 3 , σ = - 2 δ ln(R), (4)
where we set the reflection coefficient of outgoing waves to R = 10 -6 . The hyperbolic absorbing function is attractive for the following reasons: i) the resulting PML is exact at the continuous level, since F (r 0 + δ) = 0; ii) the formulation is parameter-free. Polynomial profiles such as the cubic one are parameter dependent, but the reflection coefficient R can be made arbitrarily small at the end layer. At the continuous level it is clear that both choices are satisfactory: the wavefield u s (r) is perfectly matched and decays to zero at r = r 0 + δ.

ℑ( u) ℑ(d) (b) k = π/64

Polynomial approximation of the decay function

At the discrete level, a thorough analysis is necessary to determine whether the solution u s within the PML can be accurately interpolated. We focus on polynomial discretization methods since they cover most of the numerical schemes used in this context. From equation (3) it is clear that u s (r) can be well-represented by a polynomial basis. However the discretization error linked to d(r) requires a more intricate analysis. When kr → ∞, the hyperbolic profile σ u (r) has the nice property to make the decay function linear in the layer. We indeed have

F u (r) = r r0 σ u (s) ds = -ln(1 -ξ), ξ = r -r 0 δ , and
lim kr→∞ d u (r) = (1 -ξ),
where we have introduced ξ(r) ∈ [0, 1) the scaled coordinate in the layer, and indexed the decay d u according to σ u . As a result, a simple linear function is able to exactly interpolate the decay. In this regime, the computational error is dominated by the discretization method and not by the domain truncation method. We call such a linear decay to be an optimal decay. For the cubic profile, we expect the decay d c (r) to be a smooth function at high frequencies.

In the low frequency regime, as kr → 0, the denominator of the decay function 1 + F (r) ıkr function may take large values, resulting in an abrupt decay of the transformed wavefield u, as shown in Figure 1b. Large interpolation errors are expected. To confirm this intuition, we discretize the PML domain Ω pml = [r 0 , r 0 + δ] with finite elements and compute the best polynomial approximation of the decay function in the L 2 -norm, also called the projection error. The relative projection error reads

E proj (%) = 100 ∥Pd(r) -d(r)∥ L 2 (Ω pml ) ∥d(r)∥ L 2 (Ω pml ) , Ω pml = [r 0 , r 0 + δ],
where Pd(r) = N j=1 d j ϕ j (r) is the projection of decay function on the polynomial basis {ϕ j } 1≤j≤N . We use the one-dimensional integrated Legendre polynomial basis or order p > 0, and solve the least-squares 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -6

10 -2 10 2 k E proj (%) F u (r) p = 1 p = 2 p = 4 p = 8
10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -6 problem using a Galerkin formulation with four elements. We set r 0 = 1 and δ = 0.1. The error is reported in Figure 2.

10 -2 10 2 k E proj (%) F c (r)
In the high frequency regime the hyperbolic profile is more easily interpolated by the polynomial basis than the cubic profile for low orders p. This behaviour is expected because the decay function associated to σ u is asymptotically linear. Nevertheless, the projection error linked to the cubic stretching decreases rapidly with p in this regime, which suggests that the decay function is relatively smooth. Overall the error remains sufficiently low, such that the discretization error associated to u s (r) will dominate the accuracy of the computation.

On the contrary, at low frequencies, the projection error is large even when employing a high order basis, which will drive the overall accuracy of the computation. Common absorbing functions do not perform well in this regime. This issue can be circumvented by either increasing the number of PML layers or relocating the PML further from the acoustic sources. However, such strategies come at the expense of an increased computational cost. In particular, for large frequency sweeps, regenerating the PML multiple times for different frequencies becomes necessary, when employing common absorbing functions.

We shall address these practical issues later in the numerical experiments, when dealing with domains of convex shape. In the next section, we shall propose new absorbing profiles with the goal to extend the applicability of the PML in the low frequency regime. Since we only modify the absorbing profile, the approach is expected to be valid for any conventional polynomial discretization method, hence covering most of existing industrial and academic implementation frameworks.

Improved absorbing functions

In this section, we derive new absorbing functions with the goal of obtaining a linear decay function of the first spherical harmonic for a wide frequency range. By doing so, we expect to reduce the polynomial projection error, hence improving the accuracy of the discrete PML. We focus the analysis on the radial component of the solution.

Optimal absorbing function in three-dimensions

We start our analysis by considering the first component of the expansion in [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF], that is the spherical Hankel function h

(2) ℓ (kr) with ℓ = 0. It is also the Green kernel of the three-dimensional Helmholtz operator. In order to obtain a decay function that is linear and independent of the frequency, we need to satisfy d(r) = (1 -ξ) according to Equation (3). In other words, we would like to choose the absorbing profile F (r) such that it satisfies

e -F (r) = (1 -ξ) 1 + F (r) ıkr , ξ = r -r 0 δ , in Ω pml . (5)
When kr → ∞, e -F (r) → (1 -ξ) and we verify that the unbounded profile F u (r) = -ln(1 -ξ) originally introduced in [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF] solves the problem. To solve equation ( 5), we need to allow F to be complex valued, which has already been suggested in many studies e.g. [START_REF] Kuzuoglu | Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers[END_REF][START_REF] Hugonin | Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization[END_REF][START_REF] Yang | A truly exact perfect absorbing layer for time-harmonic acoustic wave scattering problems[END_REF]. The exact solution can be expressed through the special Lambert function

W b F ex (r) = -ıkr + W b ıkr 1 -ξ e ikr , b ∈ Z, ( 6
)
where b is the branch of the multi-valued Lambert function [START_REF] Corless | On the Lambert W function[END_REF]. The Lambert function has a countable, infinite number of branches, and it is not clear at this stage how to select the correct branch for a given frequency. Fortunately, we can take advantage of the simpler Wright omega function w [START_REF] Corless | The Wright ω function[END_REF], satisfying W b (z) = w(ln(z) + 2ıπb), such as the optimal absorbing function is uniquely determined by

F ex (r) = -ıkr + w ln ıkr 1 -ξ + ıkr , (7)
which amounts to selecting the branch b = ⌈ℑ(ln(z))/(2π) -1/2⌉ = ⌈kr/(2π) -1/4⌉ of the Lambert function, using the principal value of the complex logarithm. This choice leads, for all frequencies, to a linear decay function in the PML for the spherical Hankel function h

(2) 0 (kr). This choice will also undoubtedly impact the damping of higher-order harmonics. For higher-order harmonics, we can readily extend the optimal absorbing function for wavefields of the form u s,ℓ (r) = e -ıkr /r ℓ+1 .We can solve again equation ( 7) exactly for a fixed radial mode ℓ ∈ N

F ex,ℓ (r) = -ıkr + (ℓ + 1) w ln ıkr ℓ + 1 + ıkr ℓ + 1 - ln(1 -ξ) ℓ + 1 . (8)
Unfortunately, this solution is not general enough to account for a linear combination of the radial harmonics.

In order to fit a linear decay that accounts for a combination of radial harmonics one would need to use generalized version of the Lambert function [START_REF] Maignan | Fleshing out the generalized Lambert W function[END_REF], which is out of scope of this study.

In short, we have obtained an ideal absorbing function for a spherical wave hitting a spherical PML boundary at normal incidence. The derivation is more of theoretical interest, since the evaluation of the Wright omega function in the complex domain is not standard and can be costly compared to usual mathematical functions. We recall that our goal is to reduce the projection error linked to the decay function and make the PML solution amenable to a polynomial discretization method in a wide frequency regime. To this end, we hereafter derive approximations of the optimal stretching profile (7), and we will show in Section 4 and 5 how this choice affects the damping of higher-order harmonics.

Approximate solutions

In this section, we derive approximations to the Wright omega function, which will lead to various choices for the absorbing function F . We start by fixing the leading behaviour for large arguments, such that we recover the expected PML performance at high frequencies. For large arguments we have the asymptotic expansion [START_REF] Corless | The Wright ω function[END_REF]

w(z) = z -ln(z) + o(1), z → ∞. ( 9
)
If we use the approximation ln(ıkr) ≪ ıkr as kr → ∞ in equation ( 7) and keep the first term of the asymptotic expansion, we recover the hyperbolic profile F u (r) ≈ -ln(1 -ξ) as a rough approximation. From this point, one may adopt different strategies to derive more accurate stretching functions. We discuss how to correct the leading behaviour either in the high or low frequency limit.

High frequency corrections

In this strategy, we write relation [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] under the form

F (r) = -ln(1 -ξ) -ln 1 + F (r) ıkr ,
and use a continuous fraction representation of the natural logarithm as kr → ∞. The first term gives F (r) ≈ -ln(1 -ξ) -F (r)/ıkr, leading to

F ∞ 1 (r) = -ıkr ln(1 -ξ) 1 + ıkr . ( 10
)
One may recognize a form of a convolutional PML [START_REF] Kuzuoglu | Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers[END_REF][START_REF] Bérenger | Application of the cfs pml to the absorption of evanescent waves in waveguides[END_REF], that we can interpret as the complex change of variables r = r + Fu(r) ık+1/r . The second term of the continuous fraction representation gives

F (r) ≈ -ln(1 -ξ) - 2F (r) 2ıkr + F (r) ,
which can be solved explicitly

F ∞ 2 (r) = -(1 + ıkr) + ln(1 -ξ) 2 -(1 + ıkr) 2 + (1 -ıkr) ln(1 -ξ) + ln 2 (1 -ξ) 4 ,
where we have chosen the minus sign for the square-root such that lim kr→∞ F ∞ 2 (r) = -ln(1 -ξ).

Low frequency corrections

In this strategy, we look for F under the form F (r) = -ln(S(r)), and further use a low frequency correction. Equation [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] becomes

S(r) = (1 -ξ) 1 - ln(S(r)) ıkr . ( 11
)
The idea is now to use an approximation of ln(S(r)) around S(r) = 1, that is around F (r) = 0, which occurs when kr → 0 and ξ → 0. The first order approximation is ln(S(r)) ≈ -1 + S(r). Solving equation [START_REF] Zheng | A perfectly matched layer approach to the nonlinear schrödinger wave equations[END_REF] gives the stretching function

F 0 1 (r) = ln 1 -ξ 1 -γξ , γ = 1 -ıkr 1 + (kr) 2 . ( 12
)
We can view this derivation as an approximation of the principal branch of the Lambert function in the low frequency limit. We first notice from ( 7) that when kr → 0 the principal branch b = 0 of the Lambert function will be selected. From the exact solution [START_REF] Teixeira | Complex space approach to perfectly matched layers: a review and some new developments[END_REF] we have e -F (r) = e ıkr e -W (z) . A first order Taylor expansion for the principal branch e ıkr e -W0(z) ≈ 1 + ıkr -W 0 (z) can be used since W 0 (z) → 0 when kr → 0 and ξ ̸ = 1, which we rewrite as F (r) ≈ 1 -e -F (r) . The substitution of this approximation in [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] gives [START_REF] Antoine | Towards perfectly matched layers for time-dependent space fractional PDEs[END_REF]. One may also remark that for the principal branch

W 0 (z) ≈ 1 -e -W0(z) ⇔ W 0 (z) z ≈ 1 -W 0 (z) ⇔ W 0 (z) ≈ z 1 + z ,
such that the approximation can be seen as a first order Padé approximation of W 0 (z) around z = 0. The next term of the continuous fraction representation of ln(S(r)) is

ln(S(r)) ≈ 2 S(r) -1 S(r) + 1 . (13)
Extending the previous analysis, it corresponds to the following approximation of the Lambert function for small arguments

W 0 (z) ≈ -1 + z 2 + 2z + 1 + z 2 2 , z → 0. ( 14
)
The substitution of the Padé approximation in [START_REF] Zheng | A perfectly matched layer approach to the nonlinear schrödinger wave equations[END_REF] gives a quadratic equation for S(r)

S(r) 2 + S(r) 1 -(1 -ξ) 1 - 2 ıkr = (1 -ξ) 1 + 2 ıkr ,
that we can solve explicitly. Among the two roots, we again select the sign such that we retrieve the hyperbolic profile in the high frequency limit. After algebraic manipulations we obtain

F 0 2 (r) = -ln a(1 -ξ) -ξ/2 + (2 a -1) (ξ -1) + (ξ/2 + a (ξ -1)) 2 , a = ı/kr. (15)
We show in Figure 3 the quality of the approximation of ( 13) and ( 14) for small arguments, which capture respectively two and four terms of the Taylor series of W 0 around 0. 13) and ( 14) of the Lambert function for small arguments.

-(1 + z/2) + 2z + (1 + z/2) 2

Comparison of the approximations

We compare the accuracy of the derived absorbing functions. We first measure the relative error of the resulting decay function with the optimal decay over a given frequency range in the maximum norm

E max = max Ω pml |d(r) -d opt (r)| max Ω pml |d opt (r)| , d opt (r) = (1 -ξ) . ( 16
)
The error is reported in Figure 4 over the range k = [10 -4 , 10 4 ]. The absorbing functions F 0 1 and F 0 2 show an increasing accuracy in the low frequency regime while F ∞ 1 and F ∞ 2 are more accurate at high frequency, which can be beneficial to improve the convergence properties of the discrete PML in the mid-frequency range. We further plot in Figure 5 the relative projection error of the decay function in the PML, as already performed in Figure 2. We note that the exact absorbing function F ex , which has been evaluated thanks to the algorithm [START_REF] Lawrence | Algorithm 917: Complex double-precision evaluation of the Wright ω function[END_REF], is as expected close to the machine precision. The maximal projection error for F 0 2 is 0.25% which is attained at k = 1, whereas for F ∞ 2 the maximal error reaches 3.6% at k = 10 -4 . The gain in accuracy when increasing the order p is limited, since d opt is a linear function. To sum up, we have improved the polynomial interpolability of the decay function. We see a notable improvement with the absorbing function F 2 0 in the low frequency range. We now apply these developments to relevant wave propagation problems.

10 0 k E max F u F ∞ 1 F 0 2 F ∞ 2 F 0 1

Assessment in academic cases

In this section, we assess the performance of the newly developed absorbing functions. For a fixed layer width δ, the PML applicability is extended to both large and short wavelengths up to roughly λ/10 4 , where λ is the typical wavelength of the problem. We recall that this is achieved only by modifying the PML absorbing profile, without parameter tuning.

10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -13 10 -5

10 3 k E proj (%) F ex (a) F 0 2 (r)
10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -13 10 -5

10 3 k E proj (%) p = 1 p = 2 p = 4 p = 8 (b) F ∞ 2 (r)

One-dimensional spherical wave

The first benchmark is a one-dimensional Helmholtz problem that represents the three-dimensional radial behaviour of the acoustic field. For a fixed mode ℓ ∈ N, the Helmholtz operator in spherical coordinates writes

H = ∂ 2 r + 2 r ∂ r + k 2 - ℓ(ℓ + 1) r 2 ,
for which we have the exact solution u ex (r) = A ℓ h

ℓ (kr). The spherical Helmholtz operator is discretized by a conformal p-FEM strategy using the one-dimensional integrated Legendre polynomial basis. The physical domain is the line Ω = [r * , r 0 ], r * > 0 and the PML domain is Ω pml = [r 0 , r 0 + δ]. We set a homogeneous Dirichlet boundary condition at r = r 0 + δ, as recommended in [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF]. The implementation is available in GmshFem [START_REF] Royer | Gmsh-Fem: An Efficient Finite Element Library Based On Gmsh[END_REF], an open source finite element library written in C++ based on Gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in pre-and postprocessing facilities[END_REF].

We discretize the domain with linear elements and denote with h the typical element size. The PML interface is at r 0 = 0.4, and the input boundary condition at r * = 0.1. We solve the discrete solution u h and report the relative L 2 -error

E L 2 (%) = 100 ∥u h -u ex ∥ L 2 (Ω) ∥u ex ∥ L 2 (Ω) , (17)
over the physical domain Ω for the three first spherical harmonics ℓ = {0, 1, 2} in Figure 1, using p-FEM orders p = {1, 2, 4} with respectively N pml = {5, 2, 1} layers. Each layer has the width of the typical element size such that δ = N pml h. In each plot, we show the performance of the absorbing functions F ex , F u , F ∞ 2 and F 0 2 . As a reference, the relative projection error of u ex by the p-FEM basis in the physical domain Ω is shown.

As expected, the hyperbolic profile F u from [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF] is effective for high frequencies, but shows large reflections in the low frequency regime. The function F ∞ 2 slightly improves this behaviour but is not effective in the low frequency regime, which is consistent with the analysis from Section 3.3. The newly introduced function F 0 2 is able, on the other hand, to target low frequencies effectively, while maintaining the accuracy of F u at high frequencies. This behaviour also holds with a finer discretization of the problem, for instance when increasing the p-FEM order. When p is large, the error is limited by the interpolation error of the decay function for F 0 2 and F ∞ 2 . High-order radial harmonics may also affect the accuracy of the discrete PML, because they include terms that behave as O(1/r ℓ+1 ) at low frequencies. The error plots for ℓ = 1 and ℓ = 2 suggest that the proposed absorbing functions, although derived for an optimal decay of the first radial harmonic, are also efficient on damping higher order harmonics.

The exact absorbing function F ex has the best performance among all the tested functions, but does not always reach the projection error. In our tests we use a classical Gauss quadrature with 2p + 1 points, which is not optimal to handle the singular integrand on the last PML element. We highlight this difficulty by recomputing the relative L 2 -error in Figure 7, using a graded mesh with 2p + 20 Gauss points in the PML layer. The graded mesh is formed by five linear elements of respective size h, h/2, h/4, h/8 and h/16, leading to a total layer width δ ≈ 1.93h. We are able to improve the behaviour of F ex , which suggests that quadrature (%)

10 -3 10 -1 10 1 10 3 k 10 -3 10 -1 10 1 10 3 k Figure 6: Relative L 2 -error (%) in Ω with h = 2.5 × 10 -3 . Legend: Fu ( ), F 0 2 ( ), F ∞ 2 ( ), Fex ( ), E proj ( ).
errors are also a limiting factor. We observe that the behaviour of all absorbing functions could be further enhanced by designing a tailored quadrature, which is left for a future work.

Three-dimensional point source radiation

We now assess the proposed absorbing functions in a three-dimensional situation. We consider the acoustic radiation of point sources in a bounded, spherical computational domain centered at the origin Ω = B r0 (0), and set up a PML through a classical change of coordinates with r(r) = r + F (r) ık and γ r (r) = d r dr = 1 + σ(r) ık . We define the PML tensor A through

A = RDR T , D = diag
-∇ • (A∇u) -k 2 det(A)u = 0, in Ω pml . ( 18
)
The PML domain is terminated by a homogeneous Dirichlet boundary condition. We discretize again the variational formulation of this problem with a conformal H 1 -basis based on integrated Legendre polynomials. We select a variety of input loads that are set on the boundary of a smaller sphere Ω * = B r * (x * ) of radius r * , centered at x * . We impose the trace of an acoustic monopole or an acoustic dipole oriented in the x-direction.

The boundary condition reads

∂u ∂n = ∂f m,d ∂r , on ∂Ω * , (19)
where the loads for the monopole and dipoles are, respectively

f m (r) = e -ık∥x-x * ∥ ∥x -x * ∥ , f d (r) = -cos(θ) e -ık∥x-x * ∥ ∥x -x * ∥ ık + 1 ∥x -x * ∥ . ( 20
)
For the computations, we mesh a volume domain defined between two spheres of radii r * and r 0 . A spherical PML layer of width δ = N pml h is extruded from r 0 . We use second order tetrahedral elements with a typical element size h, and fix the basis function order to p = 2. The quadrature is a tensorized Gauss-Legendre rule of order 2p + 2. We report in Figure 8 the relative L 2 -error with the exact solution in the physical domain Ω for different absorbing profiles as a function of the frequency, for the two load cases centered at the origin. We also plot the relative projection error by the polynomial basis in the physical domain. We observe a drastic improvement of the accuracy in the low frequency regime, in particular for the function F 0 2 . For the dipole source the improvement of the F 0 2 function also leads an improvement, which is however less pronounced due to the near field contribution. In the next experiment, we fix the frequency to k = 0.1 and look at the accuracy of the absorbing function F 0 2 for different sources locations x * . The inner sphere is taken of radius r * = 0.15 and the source is moved along the x-direction. The relative L 2 -errors are reported in Figure 9. Although the error increases when x * is close to the PML interface, the function F 0 2 seems relatively robust with respect to the source location. For completeness, we assess the convergence of the absorbing functions with the number of layers N pml at k = 0.1. As the frequency decreases, the number of layers required to reach the projection error increases, such that a reasonable accuracy for the usual function F u comes at the price of a large computational cost.

To sum up, the proposed function F 0 2 can reach a reliable engineering accuracy in a large frequency range with a classical finite element discretization method. It greatly improves the state-of-the-art usage of F u from [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF] by allowing a small PML width compared to the wavelength. Our experiments correspond to a PML width of approximately λ/10 6 for the lowest simulated frequency, which is several orders of magnitude below existing recommendations. In the following, we select the function F 0 2 given in Equation [START_REF] Johnson | Notes on Perfectly Matched Layers (PMLs)[END_REF] to conduct experiments in an industrial setting.

Industrial applications

As discussed is Section 3, the newly developed function results in a linear decay of the first spherical harmonic inside the PML. Since the first spherical harmonic typically dominates the behavior of low-frequency solutions, the use of the F 2 0 function is expected to improve the absorbing performance of the discrete PML at low frequencies. The proposed absorbing function is included into the commercial finite element software Simcenter 3D Acoustics [START_REF]Siemens Industry Software Simcenter Nastran user's guide[END_REF]. All computations are performed on a DELL Precision 7560 laptop with 64 Gb of RAM.

Low frequency correction for arbitrary convex shapes

In the following, the Automatically Matched Layer (AML) implementation, introduced by Beriot et al. [START_REF] Bériot | An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape[END_REF] is used. In this specific PML realization, a convex envelope surrounding the obstacle is generated using the quickhull algorithm [START_REF] Barber | The quickhull algorithm for convex hulls[END_REF] and is scaled outward by a user-defined distance δ o that dictates the size of computational domain. The PML domain is then defined by extruding the exterior boundary of width δ = N pml h with N pml layers of typical element size h. This allows to closely surround the scatterer, thus reducing the amount of unnecessary white space to discretize around the object. The complex stretching in the AML is defined from the geometrical information of the exterior surface through a modification of the Jacobian matrix, such that only the stretching function F remains to be chosen. The complex stretching is performed along the normal direction to the PML interface Γ pml , where any point x ∈ Ω pml writes

x = x 0 + δξn, x 0 ∈ Γ pml , (21)
and ξ now represents the first component of a local orthogonal coordinate system attached to Γ pml .

In the previous sections we have assumed the PML stretching to be along the radial direction only, in order to match the general solution of the exterior Helmholtz problem. This assumption does not hold anymore for generic convex shapes, and we choose to approximate the quantity kr in the stretching functions by a constant characteristic length kr ≈ kr eq , r eq = S 4π ,

where S is the surface area of Γ pml . This empirical strategy allows to extend the applicability of F 2 0 to convex boundary shapes, which in addition simplifies its implementation. Applied to the spherical PML case from Section 4.2, this strategy gives close results to the original approach that uses r = r 0 + δξ.

An efficient frequency sweep

In an industrial context, it is common to solve a large frequency sweep for time-harmonic problems. Such a sweep can be performed efficiently on a single mesh following the adaptive order strategy described in [START_REF] Bériot | Efficient implementation of high-order finite elements for Helmholtz problems[END_REF], referred to as finite element with adaptive order (FEMAO). Based on the running frequency, the method automatically selects the order of the basis functions across the mesh from a user-defined target accuracy E T . A one-dimensional a priori error indicator assigns the polynomial order of each element based on its size, allowing to sweep over the frequency range with a single mesh. Moreover, the use of a hierarchic H1 -conformal basis allows to perform the assembly of the finite element matrices only once for the highest polynomial order, before running the frequency loop.

The method presents a challenge when employed with a PML in the low frequency regime, because empirical guidelines e.g. [START_REF] Oskooi | MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method[END_REF][START_REF]QuickWave 2022 user's guide[END_REF][START_REF]Siemens Industry Software Simcenter Nastran user's guide[END_REF] either require the PML to be positioned by at least half a wavelength distance from the acoustic sources, or to increase the number of PML layers. Thanks to the enhancement of the absorbing functions from Section 3, we have the potential to guarantee a full engineering accuracy over a large frequency range using a single mesh. Once the mesh is generated, the remaining computational cost is the resolution of a series of linear systems at various frequencies, which is an embarrassingly parallel operation. This methodology offers a versatile solution for addressing industrial problems more efficiently.

Acoustic radiation of an underwater propeller

The first benchmark consists in the acoustic radiation of a 3-bladed propeller subjected to point forces at its blade tips. The scenario is challenging due to the propeller's concave shape and the nature of the excitation, which results in waves that propagate nearly tangentially to the AML surface. An underwater environment is considered, with a speed of sound c 0 = 1422m/s.

Figure 11 shows the original propeller CAD 1 and the triangular surface mesh. The characteristic edge length is h = 30 mm. An approximate convex hull is computed, surrounding the propeller, using the quickhull algorithm. The hull is then scaled outward by a length δ o = 2h and a volume mesh composed of first order tetrahedral elements is generated between the original propeller surface and the scaled hull surface, see Figure 11. The volume mesh totalizes 26 567 elements and 41 520 nodes. For the PML, five volume layers are extruded automatically from the exterior boundary using linear prisms elements. The total thickness of the layer is δ = N pml h with N pml = 5. As a reference solution, we use a spherical PML equipped with the hyperbolic absorbing profile F u . The reference interior domain is a sphere of large radius r 0 = λ/2, ensuring that the minimum distance between the propeller and the PML interface is half a wavelength from the excitation frequency. We cover simulations spanning a frequency range from 1 to 10 000 Hz, which corresponds to a wavelength range λ from respectively 1422m to 0.142m. We use five distinct meshes to compute the reference solutions, see Table 1. The target accuracy for guiding the selection of the shape function orders is set at E T = 5%.

Table 1 reports the simulation parameters and computational resources for the reference and test simulations. With a low frequency absorbing function, it is possible to use a fixed mesh for all the frequencies, 1: Simulation data for the propeller case. For the reference case, a mesh is generated at each frequency such that δo = λ/2, while a fixed mesh (Figure 11) is used for the test case with δo = 2h. The target error for the order adaptivity is E T = 5%.

hence reducing the computational and modeling cost. For the test simulation at 1 Hz, the AML is positioned over 10 000 times closer to the propeller than the recommended half-wavelength distance. Compared to the reference simulation, the memory requirement for an in-core factorization step is reduced by a factor 4. The adaptive order algorithm employs basis functions of polynomial order 1 up to 1000 Hz and next employs polynomial orders up to 3 for the simulation at 10 000 Hz, which increases the factorization memory. It is worth noting that for higher frequencies the factorization cost increases as a cubic function of the frequency, see [START_REF] Bériot | Efficient implementation of high-order finite elements for Helmholtz problems[END_REF] for more details on the adaptive order strategy. Figure 12 compares the L 2 relative error on the interface Γ pml between the reference and the test simulations. The error is computed by a least-squares average of the nodal values on Γ pml . The test simulations were ran with the F 2 0 and F u functions. The F 2 0 function provides excellent accuracy and reaches the 5% target error throughout the full frequency span on the fixed mesh. In contrast, the state-of-the-art F u absorbing function only provides an acceptable accuracy at 1000 Hz and 10000 Hz on this mesh. To summarize, the function F 2 0 coupled with an adaptive order strategy provides an interesting way to circumvent the traditional far-field PML guidelines. The modeling workflow is greatly facilitated, as a single mesh can be used for the entire frequency sweep.

Additional benchmarks

To further validate the developments performed in this article, we examine the performance of the methodology in additional scenarios, where we compare the performance of the newly proposed function F 2 0 against F u from [START_REF] Bermúdez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF]. In each case, first-order shape functions and linear elements are employed, with N pml = 5 layers such that δ = 5h. Following Section 5.3, reference solutions were established by employing a spherical PML positioned at δ 0 = λ/2 from the acoustic sources using the F u absorbing function. Four distinct models are tested, shown in Figure 13.

1. The shark submarine case 2 , which is challenging due to its elongated domain. The acoustic medium 2 CAD available online, at https://grabcad.com/library/submarine-shark-1 is water, c 0 = 1422 m/s. The model simulates the scattering of the submarine by a plane wave. 2. The tire model, which simulates the sound radiated from a tire travelling at constant speed, with the exterior tire surface subjected to an imposed velocity pattern. The acoustic medium is air: c 0 = 340 m/s. 3. The dome-shaped cavity model. The model is a spherical domain with a cavity in the form of a dome.

The excitation consists of an imposed pressure field of 1 Pa on the curved part of the cavity surface, in gray in Figure 13. The acoustic medium is air. 4. The loudspeaker model. The excitation is an imposed unit normal velocity on the bass reflexes, highlighted in orange in Figure 13. The acoustic medium is air.

Table 2 compares the performance of F 2 0 with F u in a low frequency excitation of 1 Hz for all models. The F 2 0 profile consistently shows an error below the target accuracy of 5% for various excitation and geometries, while F u is not accurate in this setting. Table 2: Additional industrial benchmarks. The relative L 2 -error on Γ pml is reported at f = 1 Hz. The characteristic edge length h and the ratio δo/λ are reported for each model. A reference is generated using a spherical PML equipped with Fu positioned at δo = λ/2. The additional tests in this subsection further underscore the benefits of the F 2 0 function for practical industrial acoustic simulations. The improved accuracy at low frequencies, when the PML interface is positioned in the near-field, circumvents the need for extensive domain sizes dictated by far-field PML guidelines. These results reinforce the conclusions drawn in Section 5.3 regarding the potential of the newly proposed absorbing function for cost and time savings in industrial applications.

Limitations and conclusion

In this article, we examined a novel methodology to derive perfectly matched layer absorbing functions for three-dimensional exterior Helmholtz problems. The core of our development relies on enforcing a linear decay of the wavefield in the layer, based on the analysis of the fundamental solutions of the Helmholtz equation. We found an exact absorbing profile involving the complex valued Lambert function, proposed and tested various approximations. The absorbing functions are able to enhance the low frequency behaviour while preserving the usual accuracy at high frequency. We showed that the approximation F 2 0 allows the PML to be placed in the extreme near field (λ/10 4 ) for sources of moderate complexity, resulting in a gain in computational and modeling time. The function is parameter-free and can be incorporated into existing finite element PML implementation.

We believe these ideas could be adapted to fundamental solutions of other wave propagation problems in the time-harmonic regime such as Maxwell or Navier equations. Due to the unbounded nature of the absorbing profiles, our developments are limited by quadrature errors which prevents to reach high-order accuracy, which could be improved in the future.

  ℓ is the spherical Hankel function of the second kind indexed by the mode ℓ ∈ N, and C mℓ ∈ C are modal amplitudes. The first terms of the spherical Hankel functions write h (2)

Figure 1 :

 1 Figure 1: Real and imaginary parts of us(r) and the decay function d(r) inside a PML layer of width δ = 0.1 set at r 0 = 1

Figure 2 :

 2 Figure 2: Relative L 2 -projection error (in %) of the decay function by the one-dimensional integrated Legendre polynomial basis, for a PML layer of width δ = 0.1 set at r 0 = 1, discretized by 4 uniformly spaced elements.

Figure 3 :

 3 Figure 3: Comparison of the approximations (13) and (14) of the Lambert function for small arguments.

10 - 4

 4 10 -3 10 -2 10 -1 10 0

Figure 4 :

 4 Figure 4: Relative error in the maximum norm of different absorbing profiles as a function of the frequency, for a PML layer of width δ = 0.1 set at r 0 = 1.

Figure 5 :

 5 Figure 5: Relative L 2 projection error (in %) of the decay function by the one-dimensional integrated Legendre polynomial basis, for a PML layer of width δ = 0.1 set at r 0 = 1, discretized by 4 uniformly spaced linear elements.

N pml = 5 p = 1 10

 1

Figure 7 :

 7 Figure7: Relative L 2 -error (%) in Ω for ℓ = 0 and h = 2.5 × 10 -3 , with a graded mesh in Ω pml and 2p + 20 Gauss-Legendre quadrature points. We refer to Figure6for the legend.

r 2 r 2

 2 γ r , γ r , γ r , R =   sin(θ) cos(φ) cos(θ) cos(φ) -sin(φ) sin(θ) cos(φ) cos(θ) sin(φ) cos(φ) cos(θ) -sin(θ) 0   with R being the rotation matrix in spherical coordinates. The transformed Helmholtz equation in the PML domain reads

10 - 4

 4 10 -3 10 -2 10 -1 10 0

Figure 8 :

 8 Figure 8: Relative L 2 -error for different absorbing functions for a centered Monopole and x-Dipole source. Simulation parameters: N pml = 2, p = 2, h = 0.075, r * = 0.7, r 0 = 1.

Figure 9 :

 9 Figure9: Relative L 2 -error for the functions Fu and F 0 2 for a moving source load at x * = (x * , 0, 0). Simulation parameters: N pml = 2, p = 2, h = 0.075, r * = 0.15, r 0 = 1. We refer to Figure8for the legend.

Figure 10 :

 10 Figure 10: Relative L 2 -error for a centered monopole load when increasing N pml . Simulation parameters: p = 2, h = 0.075, r * = 0.7, r 0 = 1. We refer to Figure 8 for the legend.

Figure 11 :

 11 Figure 11: Left: original CAD model of the propeller. Right: surface and volume mesh with h = 30 mm. The height of the propeller is h = 260 mm and the distance between the blade tip and axis of rotation is around h = 315 mm. The generated convex exterior surface is positioned at a distance 2h from the propeller.

Figure 12 :

 12 Figure 12: Relative L 2 -error computed on the interface Γ pml for the propeller test case. The mesh is shown in Figure 11.

Figure 13 :

 13 Figure 13: Industrial benchmarks. Top left: shark submarine model. Top right: tire model. Bottom left: dome-shaped cavity model. Bottom right: loudspeaker model.

CAD available online, at https://grabcad.com/library/boat-propeller-23

https://gitlab.onelab.info/gmsh/fem/-/tree/lowfreq_PML

Acknowledgements

The authors would like to thank Prof. Gwénaël Gabard for his suggestions on Padé approximation and the use of Lambert functions as well as Srijan Kumar Mukhopadhyay for his help in setting up the industrial simulation models.

Reproducibility

The source code to reproduce the simulation data from Section 4 is available online 3 in the folder examples/helmholtz1d for Section 4.1 and examples/helmholtz3d/PointSourceInSphere for Section 4.2.


Dr Kossi Tepe
email: kossi.tepe@yahoo.fr

Dr Yann Verchier
email: yann.verchier@utt.fr

Prof Yetongnon Kokou
email: kokou.yetongnon@gmail.com

The challenges of massification in higher education in Africa

Keywords: Massification, videoconferencing, smart classroom, hybridization, digital 1-Introduction

Like many developing countries, Togo faces the challenge of massification in higher education resulting from a large increase in the number of students enrolled in its public universities. Encouraged by the public authorities, with the support of the United Nations and Unesco, the number of students to be trained continues to grow to provide the country with qualified professionals and meet its socio-economic needs. The number of students in large groups (over 3,000 in some courses) raises issues of training quality and equity (availability of resources, reproducibility of content, study conditions, access to digital solutions, etc.). Access to this type of training requires special training conditions and infrastructures that are not always available in developing countries.

This article presents a qualitative study carried out with undergraduate students and teachers at the University of Lomé concerning teaching and learning conditions in large groups and a critical analysis of the solutions implemented by the university. This work can be transposed to other African countries with similar needs and will open the way to a solution analogous to intelligent classrooms for face-to-face courses.

2 Demand for quality university education is growing rapidly in developing countries. Following the implementation of the United Nations' Sustainable Development Goals (SDGs) and the involvement of UNESCO (Wikipedia), (Tepe K., Yetongnon K.,2018), efforts are being made in almost all countries to provide access to quality university education for populations. While secondary education is no longer a major challenge today in most countries, university-level training poses many challenges for developing countries concerned with training senior executives capable of tackling sustainable development problems and meeting the requirements of financial institutions such as the African Development Bank (ADB), the International Monetary Fund (IMF) and others (Pierre-Jean [START_REF] Loiret | L'université virtuelle africaine, l'enseignement à distance en trompe-l'oeil? Lavoisier[END_REF].

To meet this demand, almost all African countries have created public universities since the 1970s (lecames.org, 2018); but their numbers remain insufficient. This inadequacy has encouraged the emergence of private higher education institutions in several countries. However, the costs of these private institutions are very high in relation to the socio-economic profiles of students. In addition, they suffer from many criticisms linked to unsuitable teaching content and teaching teams' competence level.

The above factors explain the ever-increasing demand for training at African public universities. Some university courses are more in demand than others, creating large groups (3,000 students or more) in teaching some units. As a result, public universities are struggling to manage this high demand with the limited resources at their disposal.

Hosting large groups of undergraduates at public universities requires special conditions and adaptations (in terms of space and time) to ensure high-quality supervision of these students. The resources required by the teaching teams must not only be substantial but must also include additional technical resources to cope with this high demand for university training. This is the only way to ensure that training remains of the highest quality, meets the country's needs, and is recognized internationally.

Managing large groups is becoming a major challenge to which African universities must find solutions. The solutions adopted by several universities include building large classrooms for face-to-face courses and offering courses online for distance learning. These solutions are inadequate, however, as the capacity of classrooms cannot be increased indefinitely without posing new problems for audience management. On the other hand, distance learning poses new problems from the point of view of both students and teachers. These problems must be carefully studied to find appropriate solutions (access to resources, motivation, student monitoring, and certification of prior learning) [START_REF] Jézégou | La formation à distance: enjeux, perspectives et limites de la formation à distance[END_REF].

Given the high demand for training, it is necessary to provide African public universities with adequate infrastructure for hosting and managing large groups. In this article, we analyze students' perceptions of largegroup learning activities and the hybrid model of large-group management, which combines the advantages of both face-to-face and distance learning.

State of the art

Information and communication technologies have fostered the emergence and development of pedagogical innovations that affect methods of disseminating or acquiring knowledge, the types and size of target audiences, and the costs of education. Several large-scale international and national projects have been proposed to improve the capacity of African universities to meet the growing demand for higher education. These projects, generally based on ICT tools, aim to overcome the lack of infrastructure, teaching resources, and qualified teachers in certain fields. Most of these projects have not produced the expected results, and new pedagogical methodologies for mass training are being sought all over the world [START_REF] Jézégou | La formation à distance: enjeux, perspectives et limites de la formation à distance[END_REF]; [START_REF] Savoie-Zajc | Research in education[END_REF]; [START_REF] Mitchell | Online Courses and Online Teaching Strategies in Higher Education[END_REF]; [START_REF] Arbaugh | A Study Disciplinary, Structural, and Behavioral Effect on Course Outcomes in Online MBA Courses[END_REF]. There are a growing number of mass training models centered on information and communication technologies. They include:

• MOOCs (Massive Open Online Courses) and e-learning are distance learning courses aimed at the general public and offering resources for acquiring new knowledge and developing personal or professional skills, (Haipinge and Kadhila, 2021).

• E-learning enables the delivery of training courses to a target audience who can follow the course individually or collectively (e.g., planning and scheduling courses, lab and discussion sessions with online presence of teachers). Several prestigious universities have adopted these new educational technologies to offer online, fee-based training courses equivalent to the degree courses offered by these institutions in initial training (Hone, K.S.et al., 2016); [START_REF] Wu | Continuance Intention to Use MOOCs: Integrating the Technology Acceptance Model (TAM) and Task Technology Fit (TTF) Model[END_REF](Margaryan, A., et al., 2015). As part of the United Nations' ODD4 initiative, the International Institute of Online Education uses new technologies to provide learning platforms for higher education teachers. IIOE (www.IIOE.org) is a global initiative created by UNESCO-ICHEI in collaboration with higher education partner institutions in China and other developing countries in Asia-Pacific and Africa, as well as corporate partners. It aims to build the capacity of the partner institutions and expand access to quality higher education. The target audience for MOOCs and elearning may include professionals or self-taught individuals who can tailor courses to suit their

3-Study scope

TOGO has two main public universities, the University of Lomé and the University of Kara. For several years now, these two universities have been experiencing an ever-increasing demand for access to university training (post-baccalaureate) by first-time learners from secondary schools. This demand is due to the need to achieve sustainable development in line with the United Nations SDGs (SDG4), (Tepe K., Yetongnon K., 2018), (Pierre-Jean [START_REF] Loiret | L'université virtuelle africaine, l'enseignement à distance en trompe-l'oeil? Lavoisier[END_REF] and an unavoidable imperative to train senior executives in the best conditions for quality teaching. This situation is a constant concern for university managers who must deal with numerous organization and quality problems in managing large groups. The example of the University of Lomé (UL) will be used to illustrate the motivation for our study and proposed solution.

The The authorities at the University of Lomé have implemented several measures to facilitate teaching the large groups. They are based on increasing the material infrastructures and human resources, mainly:

-Constructing several large 1000 and 1500-seat amphitheaters -Setting up a wireless campus network to bring together the universities of Lomé and Kara, and hospital centers to share resources. Students at the different sites can access the Internet via a single account.

-Organization courses in parallel sessions for teaching units with very high demand -Hiring more teachers for the repetition of courses (parallel sessions). This requires harmonizing the content followed by different audiences for the same curricula.

-Setting up work and rest areas on campus to facilitate mobile work with access to the campus network.

-Putting courses online enabled not only the continuity of students' personal work, but also the -Scheduling of hybrid (distance and face-to-face) or 100% distance teaching during the pandemic.

The initial feedback from stakeholders on the measures on training at the University of Lomé (students, lecturers, technical and administrative staff, etc.) shows that the results are not satisfactory and that more needs to be done. In some teaching units, the current 1000 and 1500-seat lecture rooms do not have enough seats for all the students, who constantly compete for the available places. This situation has led to unrest and anarchic occupation of the premises. Some students have to reserve or occupy available seats in the lecture hall hours before classes start. These disruptions are more pronounced when two classes are scheduled in the same room at short intervals. Moreover, in some large lecture theaters, students have difficulty following lectures due to technical faults or poor use of teaching aids (microphones, sound systems and video projection).

4-Study methodology

To find solutions for managing large groups of students, a study was carried out to gather the opinions of the various stakeholders at the University of Lomé. The aim was to analyze the interest and effectiveness of the current solutions, as well as the possibility of considering other, more suitable solutions using ICT tools. The study adopted a mixed-method approach, combining open interviews and questionnaires designed to obtain quantitative results. It consists of :

• Questionnaires with open and closed questions for students and teachers.

•

Data collection through open interviews. These exchanges aim to collect additional information that teachers and students would not have provided in the survey forms. It should be noted that at the University of Lomé, the status of lecturers does not always allow for free expression, thus, some of the results obtained from the questionnaire must be supplemented by anonymous interviews.

The teacher surveys aimed to find out how they felt about teaching large groups (interaction, fair treatment, working comfort, etc.) and to understand how they had invested in the various distance learning solutions implemented during COVID period. It should be noted that, as the profile of the teachers varied widely (tenured, part-time, industrialists, etc.), the questionnaire began with a set of questions designed to define the profile of each respondent.

Student surveys focused mainly on the transition from high school to university regarding working conditions (group size, relationship with the teacher, personalized student support, etc.). An important part of the questionnaire focused on their connection tools (network quality, computer equipment for taking courses, etc.)

and their ability to have a suitable workspace when taking distance learning courses.

The questionnaires were sent to 1,800 students in the bachelor's program, the majority of whom were in large groups, and to 106 teachers who made lectures in various large groups. Eighty-six responses were received from teachers and 1,702 from students. The results are presented and interpreted in the following section. Seventy open interviews were conducted with teachers from various teaching teams involved in the major bachelor's and master's degree groups.

5-Results and interpretation

Teachers survey

The 86 teachers who took part in the survey agreed that an effective solution needs to be found for managing large groups of students in public universities. 58% of the survey respondents suggested dividing large groups into smaller ones for lectures. In comparison, 16.7% suggested finding a solution to relocate online courses entirely. The same percentage thought that a solution that offered the same courses online and face-to-face, to enable those who couldn't access lecture halls to take them online, would be more suitable. Informal discussions with lecturers proposing a solution for making courses available online raise the issue of technological solutions for protecting access to copies of courses and authorizing only students enrolled in these courses to listen to and view the sessions. They also raise problems of mastery of the new ICT tools that enable distance courses to be run, as noted elsewhere by the studies of (Erstad and al.,2021) and which has even led to the proposal of a basic technological skills test tool in teacher recruitment in Belgium by the work of (Tondeur et al.,2017).

50% of those surveyed thought that a reasonable size for a large group would be 50 students, 16.7% thought that a large group would contain 100 students, the same percentage thought that a large group could contain up to 300 students and only 8.3 thought that a large group could even contain up to 500 students.

With maximum sound comfort, 16.7% of opinions are divided in favor of the best possible supervision of students of a size of 100, 200, and 300. 50% believe that 50 students should not be exceeded. Further informal discussions with the majority of teachers led to the conclusion that, for practical purposes and skills assessment, the size of student groups that can be easily managed should not exceed 50 learners for reasons of material organization.

50% of the survey estimates that in one day, for efficiency in teaching the same course for 2 hours, only 2 groups will be appropriate, 41.7% estimates that up to 3 groups of students can be taught in one day. No survey expression for 4 groups in one day. During informal discussions, a majority of teachers who had already repeated the same lessons in different sessions felt that the content of the lessons was never identical. Indeed, students' questions could lead to other developments in some sessions, and sometimes fatigue or good humor can also lead to not faithfully giving the same lessons in different groups. This could have a negative impact on the acquisition of skills and the assessments of the groups put together.

83.3% of the survey affirmed that the distance learning arrangements implemented during the coronavirus pandemic were ineffective. Informal discussions with teachers suggest unpreparedness for distance learning courses. Some face-to-face practices were used online, for example teachers sought to control the presence of remotely-connected students so they could dictate lectures or interrogate them using virtual classroom ICT tools such as Discord and others. They sometimes uploaded materials to Moodle for access and appointments for live explanations. Teachers also had difficulty adapting their practices to distance-learning ICT tools, as in similar cases elsewhere. [START_REF] Blume | German teachers'digital habitus and their pandemic pedagogy[END_REF] and [START_REF] Greenhow | The educational response to Covid-19 across two countries: A critical examination of initial digital pedagogy adoption[END_REF].

To effectively achieve the pedagogical objectives of large group teaching, 50% believe that face-to-face training is needed, and 50% believe that face-to-face training should be combined with distance learning (i.e. a hybrid system).

Student survey

1702 responses were collected. The selected responses provide the following information and analysis:

• The majority of students surveyed are aged between 21 and 30 (66.7%), with only 33.3% aged between 15 and 20. Most of the 88.9% surveyed are at Bachelor's level, the level for which students are in the majority, with the problems posed to public universities by the management of these large groups.

39% of respondents believe that university studies are more difficult than high school studies.

• 72.2% of respondents are in groups of over 100 students.

• 80.6% of respondents are in groups of 1,000 to 2,000 students per class for certain teaching units.

• 58.3% of those surveyed said that the problem with large groups was communication, and 16.7% felt that there were also difficulties in carrying out the learning activities proposed by the teacher.

• Half of those surveyed do not have a workspace at home to facilitate access to the distance learning course. Among those who do have workspaces, the majority feel that working conditions do not allow for the levels of concentration that could be achieved in the classroom. This was particularly noticeable during the palliative distance learning courses organized during the coronavirus pandemic, with disturbing noises coming from the students' workplaces.

• 69.4% of those surveyed did not own a computer before attending university. Most of the courses taken before university were not based on using ICT tools. Most respondents felt uncomfortable with distance learning without a teacher alongside them, as in initial training, which discourages learning and personal development in the course modules offered online. These are some of the additional responses obtained from learners in informal discussions. This is very real when we see the work done elsewhere in e-learning solutions (Hone, K.S., El Said, G.R. 2016); [START_REF] Arbaugh | A Study Disciplinary, Structural, and Behavioral Effect on Course Outcomes in Online MBA Courses[END_REF];

(Pierre-Jean [START_REF] Loiret | L'université virtuelle africaine, l'enseignement à distance en trompe-l'oeil? Lavoisier[END_REF].

• For distance learning solutions, 72.2% believe they can take courses from their smartphone. 27.8% of students surveyed thought they could not for reasons of user-friendliness and familiarity with handling smartphones. This additional information was obtained informally from students for experiments carried out during the coronavirus pandemic.

6-Conclusion

Overall, the study points to the need to find an effective solution to the problems of managing large groups in countries with the same problems as Togo, especially for undergraduate or bachelor's degree students. Solutions based exclusively on distance learning will not be appropriate, but a face-to-face solution, or one that can combine distance and face-to-face teaching, will be more suitable. We also need to find a solution that enables courses to be automatically recorded and made available as needed to students who have not been able to attend a training module in which they are legally enrolled, and or repeat teaching sessions. Indeed :

• for students, training in a virtual environment presents a greater risk of discouragement and dropping out. These courses require technological, organizational and time management skills that most new university students do not possess. Other difficulties are linked to the lack of financial means and infrastructure to access distance learning courses. These include problems of electricity, Internet access, lack of personal workspace at home, and the acquisition of computers or equivalent devices (smartphones, tablets, etc.).

• for teachers, the virtual distance learning environment does not allow effective monitoring and evaluation of teaching and learner participation. The possibility of managing several groups of students in a single day has limits for effective teaching, and the solution to be found must take this fact into account. In addition, with informal contacts, some teachers have the feeling of being dispossessed of their teaching for distance learning courses with the online availability of courses.

The lack of mastery of new ICT tools represents a major obstacle for other teachers, who may find it difficult to adapt classroom teaching to distance learning models.

• The solutions to be envisaged must be similar to intelligent classroom solutions [START_REF] Cisse | Towards the transformation of the university into an intelligent campus[END_REF],

where teachers and learners must use ICT tools and innovative technologies at lower cost to facilitate course accessibility. The implementation of lessons with the possibility of reproducing the lessons delivered or replaying the same content without further effort by teachers with the combination of audio and video supports as proposed in education by [START_REF] Nagro | Evaluating the evidence base of video analysis: A special education teacher development tool[END_REF], [START_REF] Pouta | Student teachers' and experienced teachers' professional vision of students' rational number concept understanding[END_REF](Sommerhoff et al., 2022) would be more appropriate.

  management of distance courses by teachers. The creation of the online course platform and the training of the various users (students, teachers and technical staff) were stepped up, especially in the context of the COVID-19 pandemic.

  business model is based on reduced public funding and greater financial participation by students. The project has not been as successful as expected, and none of the participating universities have adopted its video broadcasting principle (www.campus-teranga.com). In a similar vein, the CVA project (African Virtual Campus piloted by UNESCO and its partners), launched in November 2008, aims to bring together 5 West African countries (Benin, Cape Verde, Côte d'Ivoire, Gambia and Senegal) to provide the sub-region with a common program through a network of online production and teaching centers in West Africa (with a precise model for copyright and intellectual property). This project also failed to deliver the expected results (PatrickVigliano,2009). UoPeople (University of people) (www.uopeople.edu) is a private virtual university set up in 2009 in partnership with several American and Canadian universities, offering a limited number of bachelor's and master's degrees in business, health science, education, and other fields. The teaching methodology is based on online course delivery to groups of 20 to 30 an education or make up for shortcomings in given subjects, without any training costs. The quality of the trainers and the accuracy of the content are not guaranteed. The target audience is general, global, and made up of any Internet user looking for educational videos on various academic or non-academic subjects.

		availability. Examples of popular MOOCs include Coursera (www.coursera.org), Stanford
	•	(www.Stanford.edu), and MIT (www.mit.edu). Some courses lead to professional certificates, such Videoconference-based training gained in popularity during the COVID 19 health crisis, and
		as enabled institutions around the world to maintain an acceptable level of training programming and those issued by ORACLE (www.oracle.com), CISCO (www.cisco.com),
		Microsoft(www.microsoft.com), IIOE, etc. delivery. Video Conferencing tools (ZOOM, TEAMS, MEET, WEBEX etc.) have made it possible
	• Virtual universities allow large groups to access high-quality university training at low cost. At to move course programming into a virtual space that learners can access collectively from
		the international level, several large-scale projects were proposed in the 90s, notably the African different locations. These solutions have met with some success but require further study and
		Virtual University (AVU) project, initiated by the World Bank and the ADB (Pierre-Jean Loiret, analysis to determine whether the level of teaching and assessment is comparable to that of
		2008). The goal is to transfer technology from universities in northern countries (e.g. University of traditional training.

Galway, Virginia Polytechnic University, University of Toulouse, University of Brussels) to french-speaking (Benin, Burundi, Mauritania, Niger, Senegal) and english-speaking (Ethiopia, Ghana, Kenya, Uganda, Tanzania, Zimbabwe) African higher education establishments. The pedagogical model is based on creating learning centers in African institutions, from which learners can follow a selection of courses given in the partner institutions via the Internet. The AVU's students, assisted by an instructor. Registration fees are virtually free, and the resources required

(books, videos, recordings, etc.)

for the courses are provided via the OER (Open Educational Resources, www.oercommons.org) and unesco.org (www.unesco.org) sharing platforms.

At the national level, several projects have been proposed, including the "Universités Virtuelles du Sénégal (UVS) (www.uvs.sn)", Zimbabwe ZOU (Zimbabwe Open University, www.zou.ac.zw), Pakistan (Virtual University of Pakistan, www.vu.edu.pk), and India IGNOU (Indira Gandhi National Open University, www.study badshah.com). The UVS project, created in 2013, offers an innovative approach by considering characteristics specific to developing countries, namely inadequate infrastructure, economic, financial, sociological, and ultimately political factors

(Bichat, 2012)

. The teaching methodology is based on face-to-face and distance learning courses. The strategy adopted is to offer new learners general face-to-face courses and then move to online teaching once students have acquired the autonomy, maturity, and digital skills to follow individual or group modules online. A key component of the teaching methodology is the ENO (Espace Numérique Ouvert, www.uvs.sn/eno-deluvs/presentation-eno), which gives students access to teaching resources (course videos, digital tools, etc.).

•

Training courses based on video courses on the Internet (YouTube platform), enabling tutorial videos to be put online and shared without the objective of a global training course leading to a diploma (www.youtube.com). These courses are intended more for self-training on specific subjects, with no certification of content or validation of prior learning. Content producers tend to aim for a large number of subscriptions as a reward. It's a free, autonomous acquisition of knowledge to complete

Table 1 :

 1 Table 1 below shows a ten-year trend in first-year enrolments in the faculties of literature, economics and management, law, and science at the University of Lomé. The table highlights the uneven distribution of large numbers of students across the different disciplines, necessitating a search for appropriate solutions for managing large numbers of students, particularly for lectures, discussion groups and lab sessions. Evolution

		Year	2011-	Year 2012-2013 Year		Year	Year
	Type of faculty	2012
					2019	-	2020	-	2021-2022
					2020		2021
	Literature, Languages and	2 919		3 153	5 085		4932	5 484
	Arts
	Faculty of Economics and	2 620		3 273	5 415		7080	6 270
	Management
	Law	3 876		2 246	2 772		2828	2 921
	Faculty of Science	2 290		1 877	3 083		4019	3 347

of semester 1 large group enrolments at UL over ten academic years with some reference years (academic data of University of Lomé; 2021)