Chebyshev pseudospectral time-domain method for ...

Chebyshev pseudospectral time-domain method for simulations of outdoor sound propagation Didier Dragna

∗

Laboratoire de M´ecanique des Fluides et d’Acoustique, UMR CNRS 5509 ´ ´ Ecole Centrale de Lyon, 69134 Ecully Cedex, France

Maarten Hornikx

†

Department of Mechanical Engineering, Noise and Vibration Engineering Group Katholieke Universiteit Leuven, 3001 Heverlee, Belgium

Philippe Blanc-Benon

‡

Laboratoire de M´ecanique des Fluides et d’Acoustique, UMR CNRS 5509 ´ ´ Ecole Centrale de Lyon, 69134 Ecully Cedex, France

Franck Poisson§ SNCF, Direction de l’Innovation et de la Recherche, 75379 Paris Cedex 08, France

Roger Waxler

¶

National Center for Physical Acoustics University of Mississippi, University, Mississippi 38677, USA

Abstract Time-domain methods for solving linearized Euler equations have become a new standard in the community of outdoor sound propagation thanks to increased computing power. Indeed, these methods can account for most of the physical phenomena governing atmospheric sound propagation. One of the main issues is still that simulations of long-range propagation require a large numerical domain and a significant number of time iterations. Special attention must therefore be paid to the time integration and to the spatial derivative operator to produce accurate results at reasonable numerical cost. Several methods have been proposed to compute the spatial derivative. Among them, finitedifference techniques are the most widespread. Optimized schemes developed in the computational aeroacoustics community offer a solution. An other way to solve linearized Euler equations is to use pseudospectral methods. These methods allow to compute waves without numerical dissipation and numerical dispersion and require fewer points per wavelength than finite-difference methods. Fourier-type methods have been developed to treat propagation of acoustic waves in media with discontinuous properties. However, one of the drawback of these methods is the treatment of boundary conditions. This difficulty can be overcome by the use of the Chebyshev collocation pseudospectral method. In this work, two different pseudospectral time-domain solvers based on the Chebyshev method are considered to treat sound propagation over impedance ground surfaces. A long-range configuration is studied to validate the pseudospectral solvers. Then, the solvers are applied to model propagation of temporal signals with compact support in a moving atmosphere.

1

Introduction

Atmospheric sound propagation involves many complex physical phenomena related to the properties of the propagation medium (refraction due to temperature and wind variations in the atmospheric boundary ∗ PhD

student, [email protected]. research fellow, [email protected]. ‡ Senior Research Scientist at CNRS, [email protected]. § Research Engineer, [email protected]. ¶ Research Scientist, [email protected]. † Postdoctoral

1

layer, scattering by small-scales turbulence, ..) and to the properties of the boundaries (reflection on a impedance ground surface, effects of topography, ...). Analytical solutions exist only in simple cases. Numerical simulations are then required to study sound propagation in complex environments. Due to increasing computing power, time-domain simulations have become more and more popular in the community of outdoor sound propagation. Indeed, they allow to consider complex phenomena such as effect of a moving atmosphere on acoustic waves, diffraction by obstacles and to model moving sources [1, 2, 3]. Several numerical issues have been addressed, mainly concerning the time integration operator and the spatial derivative operator. Indeed, for long range applications, the numerical domain is generally large and a high number of time iterations is needed. Two types of numerical techniques for the computation of spatial derivatives have been mainly proposed. Finite-difference techniques are the most widespread, partly due to ease of implementation. They approximate the function u(x) by a local interpolating polynomial. The derivative of this function is then calculated from the derivative of the polynomial [4]. Optimized schemes developed in the community of computational aeroacoustics have been used sucessfully in long range propagation cases [5]. Otherwise, spectral methods can also be applied. These methods have the property to have spectral accuracy. This means that the error committed with differentiation decreases exponentially with the number of points down to the machine precision. The function u(x) is in this case expressed as a series of global smooth functions φn (x): u(x) ≈

N !

Un φn (x).

(1)

n=0

The derivative du/dx can then be computed from the derivatives of these basis functions. Among the spectral methods, pseudospectral methods are here considered. The expansion over the basis functions is forced to be equal to u(x) at some points called collocation points. Fourier series are the best choice because they can accurately calculate acoustic waves down to two points per wavelengths. However, they require signals to be spatially periodic, which is not the case for applications in outdoor sound propagation. Hornikx et al. [6] have proposed an extended Fourier pseudospectral method to model propagation of acoustic waves in media with discontinuous properties. This method has proved to be numerically efficient in particular to account for rigid boundaries. To overcome this limitation, Chebyshev polynomials could be chosen as basis functions instead of trigonometric polynomials. In this paper, pseudospectral time-domain (PSTD) solvers are developed to take into account complex boundary conditions. Firstly, a Chebyshev PSTD solver is proposed: spatial derivatives are calculated in all spatial directions by using the Chebyshev PS method. Then, an hybrid Fourier-Chebyshev PSTD solver is considered, where the Chebyshev PS method is used only for the derivative in the direction perpendicular to the ground. The time-domain impedance boundary condition proposed by Cott´e et al. [7] is implemented in both solvers to model the ground effect. In Sec. 2, the method is briefly described. The solvers are then validated with a long range configuration test-case in Sec. 3. Numerical efficiency is compared with finite-difference techniques. Finally, in Sec. 4, two cases are considered to show that meteorogical effects can be accurately taken into account.

2 2.1

PSTD method Spatial derivative operator

In this section, the pseudospectral methods used thereafter are briefly described. Fourier PS method Fourier series are the best choice for the function basis φn (See Eq. 1) when the solution is periodic. Indeed, by using this basis, acoustic wavelengths are accurately resolved down to two points per wavelengths. This limit corresponds to the Nyquist-Shannon sampling theorem. Assuming that N is even and restricting to the spatial interval [0; 2π], the solution is approximated by: [8] u(x) ≈

N !

Un exp(ikn x),

n=0

2

(2)

with kn = −N/2 + n. The coefficients Un can be found using the relation: Un =

1 2π

"

2π

u(x) exp(−ikn x)dx.

(3)

0

Calculation of this integral can be performed using a quadrature approximation: [8] Un =

N −1 1 ! u(xj ) exp(−ikn xj ) c˜n N j=0

(4)

with c˜n = 2 for n = 0 and n = N and c˜n = 1 otherwise. The collocation points xj are then xj = 2πj/N , which correspond to an equidistant grid. The derivative of u(x) at the collocation points is then given by: N −1 ! ∂u (xj ) = ikn Un exp(ikn xj ), (5) ∂x n=1 because the contributions from U0 and UN cancel out each other. Chebyshev PS method The Fourier PS method fails for non-periodic problems, due to Gibbs oscillations. Orthogonal polynomials such as Chebyshev or Legendre polynomials can be used in this case. Chebyshev polynomials, denoted by Tn , are considered here. Without loss a generality, the finite interval x ∈ [−1; 1] is now considered and the solution is approximated by: u(x) ≈

N !

Un Tn (x).

(6)

n=0

Due to weighted orthogonality of Tn , the coefficients Un are given by: Un =

2 cn π

"

1

−1

u(x)Tn (x) √ dx 1 − x2

(7)

where c0 = 2 and cn = 1 otherwise. This integral can be calculated by using a Gauss quadrature [8]: Un =

N 2 ! wj u(xj )Tn (xj ). cn π j=0

(8)

The collocation points xj are here chosen to be the Gauss-Lobatto points. Indeed, this distribution is closed to the best one and includes the end points of the interval. Boundary conditions can then be easily imposed. The Gauss-Lobatto distribution is given by : # $ πj , j = 0, ..., N, (9) xj = − cos N where the weights wj are given by: wj =

π c˜j N

(10)

with c˜j = 2 for j = 0 and j = N and c˜j = 1 otherwise. In the Chebyshev pseudospectral method, due to the Gauss-Lobatto distribution, the average grid size and the greatest grid size are respectively ∆xavg = 2/N ∆xmax = π/N ; then, wavelength down to π times the average mesh size can be calculated. If we denote x = − cos ξ, the derivative of u(x) can then be calculated by using a chain rule: N N ! ∂ξ ∂ ! 1 ∂u (x) = Un Tn (− cos ξ) = − √ nUn (−1)n sin nξ. ∂x ∂x ∂ξ n=0 1 − x2 n=0

3

(11)

At the node points xk , the derivative of u(x) can then be evaluated by: [9] $ # N ! ∂u 1 njπ for j = 1, ..., N − 1, (xj ) = − % nUn" sin ∂x N 1 − x2 n=0

(12)

j

N ! ∂u (x0 ) = − n2 Un" , ∂x n=0

(13)

N ! ∂u (xN ) = − n2 (−1)n+1 Un" , ∂x n=0

(14)

where the coefficients Un" = (−1)n Un are given by: Un" =

$ # N 2 ! 1 njπ for n = 0, ..., N. u(xj ) cos cn N j=0 c˜j N

(15)

For both methods, the computation of the coefficients and the derivative can be made through matrix multiplication or through the Fast Fourier Transform (FFT) [9]. The first approach has a numerical cost of the order of N 2 while the cost of FFT for real-valued data is of the order of N log N . Boyd [4] points out that matrix multiplication is faster for small values of N and that FFT becomes faster for N = 8 to N = 512, depending on the software and the hardware. The FFT technique is here prefered. The solver is developed in FORTRAN and uses the FFTPACK library.

2.2

Numerical implementation

Linearized Euler equations Pseudospectral methods are here used to solve the two-dimensional linearized Euler equations: ∂p + V0 · ∇p + ρ0 c2 ∇ · v = ρ0 c2 Q, ∂t ∂v + ρ0 (V0 · ∇)v + ρ0 (v · ∇)V0 + ∇p = R. ρ0 ∂t

(16) (17)

with p the acoustic pressure, v = (vx , vz ) the acoustic velocity, c the adiabatic sound speed, V0 = (V0x , V0z ) the mean flow, ρ0 the mean density, Q a mass source term and R = (Rx , Rz ) external forces vector. These equations are valid to order |V0 |/c. At the outer boundaries, a Perfectly Matched Layer (PML) is used. This PML allows the variables to have a compact support in the computational domain. This property is essential to use Fourier PS method. Following Berenger [10] and Hornikx et al. [6], the pressure is split into components parallel and perpendicular to the PML layer. The following equations are then obtained: ∂px ∂vx ρ 0 c2 Q + ∇.px V0 + ρ0 c2 + σx px = , ∂t ∂x 2 ∂vz ρ 0 c2 Q ∂pz + ∇.pz V0 + ρ0 c2 + σz pz = , ∂t ∂z 2 ∂p ∂vx + ρ0 (V0 .∇)v + ρ0 (v.∇)V0 + + σx ρ0 vx = Rx . ρ0 ∂t ∂x ∂vz ∂p + ρ0 (V0 .∇)v + ρ0 (v.∇)V0 + + σz ρ0 vz = Rz , ρ0 ∂t ∂z

(18) (19) (20) (21)

where the acoustic pressure is given by p = px + pz and where σx and σz are the PML coefficients respectively in the x-direction and in the z-direction. It has been shown by Diaz and Joly [11] that the error between the solution for the wave equation in free-field and the solution for the wave equation when a PML of finite length is used converges uniformly to zero with a spectral dependance on σ ¯ , the mean value of σ, L the length of the PML and the distance from the source. However, due to discretization, 4

numerical dispersion is introduced and σ ¯ should not be too large [12]. The coefficients of the PML are here chosen as smooth functions to avoid discontinuity at the interface of the PML layer: & x 'β & z 'β σx = σ0 , σz = σ0 . (22) L L For numerical purpose, the above equations are written in the following conservative form: ∂U ∂E ∂F + + + H = S, ∂t ∂x ∂z

(23)

where the different fluxes are given by: 1 px C B B p C z C =B C, B Bρ v C @ 0 xA

U

ρ0 vz

σx px ρ0 c2 Q/2 V0z px V0x px + ρ0 c2 vx B C C B C C B Bρ c2 Q/2C B C C BV p + ρ c2 v C σ p V0x pz z z 0 zC B 0 C B C C B 0z z C. C, H =B C, S =B C, F =B B C Bρ v.∇V C B Rx C V0z ρ0 vx V0x ρ0 vx + p C 0x + σx ρ0 vx A @ 0 @ A @ A A ρ0 v.∇V0z + σz ρ0 vz Rz V0z ρ0 vz + p V0x ρ0 vz 1

0

0

E

B B =B B B @

1

0

0

1

0

1

In a first solver called Chebyshev PSTD solver, the derivatives are computed in the x and z directions with the Chebyshev PS method. In a second solver called hybrid Fourier-Chebyshev PSTD solver, the derivative in the z-direction uses the Chebyshev PS method while in the x-direction the Fourier PS method is used. Typical mesh grids for the two solvers are represented in Fig. 1. PS Fourier-Chebyshev

PS Chebyshev z

z

x

x

Figure 1: Typical mesh grids (left) for the Chebyshev PS solver and (right) for the hybrid FourierChebyshev solver. In both cases, a multidomain Chebyshev PS method is used. The red lines represent the interfaces of the subdomains for the Chebyshev PS method.

Time-integration For the time-integration, the optimized six-stage Runge-Kutta algorithm of Bogey and Bailly [13] is used. This Runge-Kutta algorithm has been initially developed to minimize dissipation and dispersion errors for ω∆t ≤ π/2, where ω is the angular frequency and ∆t is the time step. This limit corresponds to a acoustic wave whose period is equal to four times the time step. It can be linked to the wavelength through the CFL number by the relation ω∆t = CFL k∆x. The time step is given by the relation ∆t = CFL ∆xmin /c0 , where ∆xmin is the smallest mesh grid size. Due to the Gauss-Lobatto quadrature, the ratio of the average mesh size over the smallest mesh size is equal to ∆xavg /∆xmin = 4N/π 2 . Thus, for a fixed domain size, the smallest mesh grid size increases quadratically with the number of points, unlike the Fourier PS method in which the increase is linear. This is the main drawback of the Chebyshev PS method. Ground boundary conditions Two conditions are used at the ground boundary. For a rigid ground, the normal velocity is set to zero. For ground surfaces wth a finite impedance, the time-domain impedance boundary condition (TDBC) 5

proposed by Cott´e et al. [7] is implemented. Firstly, the surface impedance Z is approximated in the frequency-domain by a rational function : Z(ω) = Z∞ +

N !

k=1

Ak , λk − jω

(24)

where λk are the poles, Ak corresponding coefficients and N the number of poles. The parameter Z∞ represents the value of Z(ω) when ω tends to infinity. With this form, a recursive convolution can be (m) used. Considering discrete time step ∆t and denoting p(m) = p(m∆t) and vn = vn (m∆t), the following TDBC is obtained: N ! (m) p(m) = Z∞ vn(m) + Ak φk , (25) k=1

where φk are called accumulators. Assuming that vn is contant over a time step, these accumulators are given by the recursive formula: (m)

φk

= vn(m)

1 − e−λk ∆t (m−1) −λk ∆t + φk e . λk

(26)

In this TDBC, N accumulators are needed, with only two storage locations per accumulator. This TDBC has already been used to study long range propagation over a flat ground with different impedance models [5, 14]. At each stage of the Runge-Kutta algorithm, the pressure is then imposed at z = 0 with the relation 25. Subdomains for Chebyshev PS method As pointed out in Sec. 2.2, the time step ensuring stability for Runge-Kutta schemes is linked to the smallest mesh grid size trough the CFL condition. For large numerical domains with the Chebyshev PS method, one then need to split the domain into subdomains to relax the time step. Information has then to be propagated at the interface of two subdomains at each stage of the Runge-Kutta algorithm. The method of characteristic variables can be used [15]. This method involves to rewrite the linearized Euler equations in the form: ∂q ∂q ∂q +A +B + Cq = D, (27) ∂t ∂x ∂z with the variable vector q = [px , pz , ρ0 cvx , ρ0 cvz , ]T and matrices A, B, C and D. To find the variables propagating for instance in the x-direction, the matrix A is diagonalized, such that A = P −1 ΛP where Λ is a diagonal matrix. The variable q¯ = P −1 q is introduced and after substitution in Eq. 27, the following system is obtained : ∂¯ q ∂¯ q ∂¯ q + A" + B" + C"q ¯ = D" , (28) ∂t ∂x ∂z where the matrices A" , B " , C " and D" depend on P and on matrices A, B, C and D. For a homogeneous atmosphere with c = c0 and for the Eq. 18-21 without source terms, the following equations are obtained: ∂ q¯1 c0 ∂ q¯3 +√ + σx q¯1 = 0, (29) ∂x 2 ∂z ∂ q¯2 c0 ∂ q¯3 − c0 +√ + σx q¯2 = 0, (30) ∂x 2 ∂z c0 ∂(¯ q1 + q¯2 ) +√ + σz q¯3 = 0, (31) ∂z 2 √ ∂ q¯3 + c0 2 + σz q¯4 = 0, (32) ∂z √ √ √ where the characteristic variables are: q¯1 = (ρ0 c0 vx + p)/ 2, q¯2 = (−ρ0 c0 vx + p)/ 2, q¯3 = 2ρ0 c0 vz and q¯4 = 2pz . The variables q¯1 and q¯2 represent waves travelling respectively in the x and −x directions. The variables q¯3 and q¯4 represent non-travelling waves in the x-direction. At each stage of the Runge-Kutta algorithm, the characteristic variables are computed from the physical variables at the interfaces of the ∂ q¯1 ∂t ∂ q¯2 ∂t ∂ q¯3 ∂t ∂ q¯4 ∂t

+ c0

6

subdomains. The physical variables are then corrected at the interfaces assuming that the outward travelling characteristic variables are calculated correctly. The non-travelling characteristic variables are averaged. For instance, if we consider an interface in the x-direction, the characteristic variables q¯1l and q¯2r are computed respectively at the left and right interfaces. Assuming the continuity of pressure in the x-direction and normal velocity at the interface, the physical variables are then updated with: √ p = (¯ q1l + q¯2r )/ 2, (33) pz = (¯ q4l + q¯4r )/4, px = p − pz ,

√ ρ0 vx = (¯ q1l − q¯2r )/( 2c0 ), √ ρ0 vz = (¯ q3l + q¯3r )/( 2c0 ).

(34) (35) (36) (37)

For the corner points in the Chebyshev PSTD solver, the average value of each variable is computed and is then imposed. A schematic view of the calculation of the characteristic variables is represented in Fig. 2.

l q¯1,x

r q¯2,x

u q¯2,z

d q¯1,z

Figure 2: Representation of the calculation of the characteristic variables at interfaces of four subdomains.

3

Long range configuration

Propagation above a finite impedance flat ground in homogeneous conditions is here considered to validate the method. The two solvers proposed in Sec. 2.2 are used.

3.1

Numerical parameters

Comparisons in the frequency-domain are realized for a receiver located at x = 400 m and z = 2 m. The numerical domain is then [-12 m; 412 m] in the x-direction and [0 m; 48 m] in the z-direction. The length of the PML is L = 2 m and the following parameters are used σ0 = 25 × 103 s-1 and β = 4. The used impedance model is the Miki model [16] of a semi-infinite layer of effective flow resistivity 100 kPa.s.m-2 . The coefficients of the approximation of the impedance model can be found in Cott´e et al. [7] under the designation “OFv1”. The source is a gaussian pulse located at xS = 0 m and zS = 2 m: ) ( (x − xS )2 + (z − zS )2 δ(t), (38) Q(x, z, t) = exp − B2 with B = 0.4 m. 7

Four simulations are performed: the number of points in the subdomains for the Chebyshev PS method is set to N = 16 and N = 32. The number of points is chosen small enough to have an acceptable time step and large enough to get spectral accuracy. For the Chebyshev PS method, the average mesh grid size is set to ∆zavg = 0.125 m; assuming that the Chebyshev PSTD solver allows to calculate acoustic wavelengths down to π times the mesh grid size, the maximum resolved frequency is then equal to 865 Hz. For the hybrid Fourier-Chebyshev PSTD solver, the average mesh grid size is ∆zavg = 0.125 m in the z-direction and the mesh grid size in the x-direction is increased with ∆x = 0.196 m. In all simulations, the CFL number is set to 1.

3.2

Results

The waveforms at a receiver located at x = 400 m and z = 2 m obtained with the differents cases are represented in Fig. 3. In all cases, the waveforms are similar and have mainly a low-frequency content. The results are now compared to an analytical solution, based on the solution of Di and Gilbert [17]. The corresponding spectra are plotted in Fig. 4. It can be seen that for low-frequencies, a very good agreement is found for the different simulations. For the Chebyshev PSTD solver with N = 16, the numerical result deviates from the analytical calculation for frequency higher than around 480 Hz; this value corresponds to a maximum resolver frequency equal to 5.7 times the mesh grid size. For the Chebyshev PSTD solver with N = 32, the deviation occurs for frequency higher than 660 Hz; this value corresponds to approximately 4 points per wavelength. A good accuracy is obtained with the hybrid Fourier Chebyshev PSTD solver for N = 16 and N = 32 up to 865 Hz. Indeed, the spectra obtained with the hybrid Fourier Chebyshev PSTD solver and with the analytical solution are almost indistinguishable. −3

x 10

PSTD theory

Pressure, Pa

4 2 0 −2

1175 1180 1185 1190 1195 Time, ms

Figure 3: Waveforms at a receiver located at x = 400 m and z = 2 m obtained with the PSTD solvers and with an analytical solution.

0

C 16 C 32 theory

−50

SPL, dB

SPL, dB

0

−100

−150 0

FC 16 FC 32 theory

−50

−100

200

400 600 Frequency, Hz

−150 0

800

200

400 600 Frequency, Hz

800

Figure 4: Sound pressure level at a receiver located at x = 400 m and z = 2 m (left) for the Chebyshev PSTD solver and (right) for the hybrid Fourier Chebyshev PSTD solver.

8

−50

0

C 32

Error (phase), dB

Error (amplitude), dB

C 16 FC 16 FC 32

−60

−100

C 16 C 32 FC 16

−120 0

FC 32 200

400 600 Frequency, Hz

−150 0

800

200

400 600 Frequency, Hz

800

Figure 5: Errors on amplitude (left) and on phase (right) obtained with the different calculations versus frequency. These results are now going to be confirmed quantatively with two estimators defined by : * * * * * φana − φnum * * |pana | − |pnum | * phase amp * * *, * = 20 log * + = 20 log * * and + * pana φana

(39)

where pana and pnum are respectively the pressure in the frequency-domain obtained analytically and with the numerical calculation and where φana and φnum are the corresponding phases. These two estimators correspond respectively to the relative error on amplitude and to the relative error on phase. They are plotted versus frequency in Fig. 5 for the different calculations. As noticed above, the amplitude error is small at low frequencies and increases rapidly with the Chebyshev PSTD solver at high frequencies. Note also that the amplitude error is similar for the calculations with the Fourier Chebyshev PSTD solver for both N = 16 and N = 32. Concerning the phase error, it can be seen that at low frequencies, the same value is obtained for the Fourier Chebyshev PSTD solver and for the Chebyshev PSTD solver. Discrepancies between the calculations with N = 16 and N = 32 could be due to the smallest time step used for N = 32, that ensures a smallest error in time integration.

3.3

Comparison of the computational costs

C N = 16 C N = 32 FC N = 16 FC N = 32

number of points 1.47 ×106 1.38 ×106 0.88 ×106 0.86 ×106

number of iterations 28311 56486 28311 56486

tCP U , s 1.23×105 2.67×105 1.09×105 1.45×105

tCP U /nbt, s 4.4 4.7 3.84 2.6

f−20 dB , Hz 490 630 814 814

Table 1: Parameters of the different calculations. In this section, the solvers are evaluated in terms of computational costs. Several parameters can then be found in Tab. 1. First, it can be seen that the hybrid Fourier Chebyshev PSTD solver requires less points than the Chebyshev PSTD solver; indeed, there are less points in the x-direction for the Fourier Chebyshev PSTD solver. The number of iterations is doubled when doubling N due to the Gauss-Lobatto quadrature. Note also that the computational time per iteration is quite constant with the number of points N . The calculation with N = 32 needs then two times much time due to the number of iterations. It seems also that in this case, the hybrid Fourier Chebyshev PSTD solver requires less computational time that the Chebyshev PSTD solver. Finally, to compare the errors obtained for the different calculations, an accuracy limit based on the criteria +amp > −20 dB is proposed. This limit corresponds to an error on 10 % for the amplitude of the pressure. It is expressed in term of the frequency, denoted as f−20 dB , for which this limit is reached. As noticed in the previous section, the Fourier Chebyshev PS method is the most favorable. 9

It has been seen on this test case that the hybrid Fourier Chebyshev PSTD solver seems promising. It will be applied in next section to model propagation of impulsive signals.

4

Two-dimensional simulations with meteorogical effects

In this section, it is shown that meteorogical effects can be taken into account with the PSTD solvers. Impulsive signals are considered. Indeed, the solvers allow to use a moving frame to reduce both memory and computational time [18]. With the Chebyshev PSTD solver, it is complicated to implement a moving frame beacuse the mesh is not uniform in the x-direction. On the contrary, a moving frame can be straighfordwardly used with the Fourier-Chebyshev PSTD solver. This solver will then be considered here. The numerical domain has 384 points in the x-direction and 25 subdomains with 32 points in the z-direction. The mesh grid size is approximately 0.1 m in the x-direction and 0.0625 m in the z-direction. The CFL number is chosen as 2.03 so that the ratio ∆x/(c0 ∆t) is an integer as large as possible. Here, this ratio is equal to 10 and thus, the grid is moved rightwards every 10 iterations. −3

1

1.5

x 10

0.5 |S(ω)|

s(t)

1 0

0.5 −0.5 −1 0

5

10 Time, ms

15

0 0

20

300

600 900 Frequency, Hz

1200

Figure 6: Source signal in the time domain (left) and source strength in the frequency domain (right). The source is a point mass source: Q(x, y, z, t) = s(t) δ(x − xS ) δ(z − zS ),

(40)

where the source signal is: s(t) = S0 sin(2πfc t) exp

(

) (t − tc )2 H(t), τ2

(41)

with H(t) the Heaviside function. The Heaviside function is needed for the causality condition. If the parameters tc and τ are well chosen, it can be dropped from the above equation. The source strength S(ω) obtained by taking the Fourier transform of s(t) is then given by: # $ # $) √ ( (ω − ωc )2 τ 2 (ω + ωc )2 τ 2 i πτ exp[i(ω − ωc )tc ] exp − − exp[i(ω + ωc )tc ] exp − , S(ω) = S0 2 4 4 with ωc = 2πfc . The term fc corresponds to the central frequency and is set to fc = 600 Hz. The amplitude S0 is here chosen as S0 = 1 Pa.m-2 .s-1 . The parameter τ governs the decay rate of the Gaussian and is set to τ = 1.4 ms. Lastly, the variable tc is set to tc = 10 ms. The source signal and the source strength are represented in Fig. 6. As pointed out by Hornikx [6], using a point source causes a discontinuity in the pressure field. However, with the extended Fourier PSTD solver, ringing effects were not discernible in the results. It is not clear that the same statement can be made with the hybrid Fourier Chebyshev PSTD solver. In particular, it is prudent not to attach the source point to a grid point located on a subdomain boundary. In the following cases, the source is located at xS = 0 m and zS = 3 m. Around 33 000 time iterations are performed so that the pulse can reach receivers located at x = 300 m. 10

4.1

Sound speed profile

In a first case, a stratified atmosphere with upward-refracting conditions is considered. The so-called bilinear profile is used: c0 c(z) = + , (42) 1 + 2z/Rc

with Rc = 1000 m. With this value of Rc , the sound speed decreases almost linearly with height up to 50 m. The Miki impedance model of a semi-infinite layer with flow resistivity 100 kPa.s.m-2 and of a rigidly backed layer of thickness 10 cm and of effective flow resistivity 10 kPa.s.m-2 are considered. Thereafter, these two impedance models will be denoted respectively as “grass” and “snow”. The set of coefficients for the approximations of the impedance models can be found in Dragna et al. [14]. The scheme of the problem is depiected in Fig. 7. z

c(z)

Receiver Source Z

x

Figure 7: Schematic of the problem The results from the numerical calculation are compared in the time-domain to an analytical solution proposed by Berry and Daigle [19]. The analytical waveforms are computed with an inverse Fourier transform: " +∞ 1 p(x, z, t) = ρ0 iωAS(ω)ˆ p(x, z, ω) exp(−iωt) dω. (43) 2π −∞ where pˆ(x, z, ω) is the solution of the equation: ∆ˆ p + k(z)ˆ p = δ(x − xS )δ(z − zS )

(44)

with k(z) = ω/c(z). It should be noted that writing the spatial dependance of a source term implemented in a numerical solver as a Dirac is purely formal. Indeed, there will be a constant factor, denoted here as A in Eq. 43 between the desired source strength and the actual source strength. This factor A depends on the grid, and has the unit of m2 for two-dimensional calculations. A preliminary calculation in free-field shows that the factor A is close to ∆x2 in this case. The waveforms obtained at a receiver located at x = 300 and z = 2 m are plotted in Fig. 8. It can be seen that the waveforms obtained analytically and numerically coincide very well. The corresponding sound pressure level is represented in Fig. 9. Over the frequency band of interest, a close agreement is obtained. For frequencies higher than 1400 Hz, deviations can be observed in both cases. Some discrepancies can also be seen for the interference pattern obtained for the impedance model called snow for frequencies around 100 Hz. However, this is mainly due to the windowing of the waveforms.

11

−4

−4

x 10

x 10 num theory

0.5 0 −0.5

num theory

1 Pressure, Pa

Pressure, Pa

1

−1

0.5 0 −0.5 −1

−1.5 880

900

920 Time, ms

−1.5 880

940

900

920 Time, ms

940

Figure 8: Waveforms obtained at a receiver located at x = 300 m and z = 2 m for the impedance models called snow (left) and grass (right). The solid line and the dashed line represent respectively the results from the numerical calculation and from the analytical solution.

num theory

−150 SPL, dB

SPL, dB

−150

−200

−250 0

num theory

−200

−250 500 1000 Frequency, Hz

1500

0

500 1000 Frequency, Hz

1500

Figure 9: Sound pressure level for a receiver located at x = 300 m and z = 2 m for the impedance models called snow (left) and grass (right). The solid line and the dashed line represent respectively the results from the numerical calculation and from the analytical solution.

4.2

Moving atmosphere

In this second case, an uniform horizontal wind velocity V0x = M c0 is considered. The Mach number M is set to 0.05. A rigid ground is used. The scheme of the problem is represented in Fig. 10. z

V0x = M c0

Source

Receiver

Z=∞

x

Figure 10: Schematic of the problem The analytical solution can be found in Ostashev et al. [1]. The waveforms obtained at a receiver located at x = 300 m and z = 2 m and the corresponding sound pressure level are plotted in Fig. 11. A very good agreement is obtained. Some discrepancies can be observed in the sound pressure level for 12

frequencies higher than 1550 Hz.

num theory

0

num theory

0

0.2 SPL, dB

Pressure, Pa

0.4

−75

−0.2 −150 −0.4 846

848

850 852 Time, ms

854

0

500 1000 Frequency, Hz

1500

Figure 11: Waveforms obtained at a receiver located at x = 300 m and z = 2 m (left) and corresponding sound pressure level (right). The full black line and the dashed red line represent respectively the results from the numerical calculation and from the analytical solution.

5

Conclusion

To solve the LEEs, pseudospectral time-domain methods (PSTD) are an alternative to finite-difference time-domain (FDTD) methods. They may be beneficial in terms of memory and computational cost. A Linearized Euler Equations solver using an extended Fourier PSTD method has been recently proposed to model propagation of acoustic waves over a ground with a real-valued impedance. Here, an extension is considered to take into account more realistic ground impedance models. To do so, the Chebyshev PS method is used because it allows to impose such a boundary condition. A time domain impedance boundary condition is then straighfordwardly implemented at the ground. Two solvers are proposed. The first one uses the Chebyshev PS method to calculate the derivatives in all directions. The second uses only the Chebyshev PS method to calculate the derivatives in the direction perpendicular to the ground. In the other directions, the Fourier PS method is used. A long range propagation test-case has been investigated in order to validate the solvers. The hybrid Fourier Chebyshev PSTD solver has been shown to be more efficient from a numerical point of view. This solver has then be applied to two cases involving meteorogical effects on sound propagation. It has been seen that a good agreement was obtained in both cases with analytical solutions. The spatial discretisation with the Chebyshev PS method is non-uniform and, the smallest mesh grid size decreases quadratically with the number of points. In explicit time integration methods, the time step is directly linked to this smallest mesh grid size, and will then also decreases quadratically with the number of points. This has two main consequences. Firstly, for classical applications, the domain has to be split into subdomains. Moreover, a large number of iterations is then needed even for moderate number of points in each subdomains. Some techniques [20] have been proposed in the literature so that the time step decreases linearly with the number of points, and will be investigated in future work.

Ackowledgements Support by CNRS and SNCF is gratefully acknowledged. This work was granted access to the HPC resources of IDRIS under the allocation 2011-022203 made by GENCI (Grand Equipement National de Calcul Intensif).

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References [1] V.E. Ostashev, D.K. Wilson, L.Liu, D.F. Aldridge, N.P. Symons, and D. Martin. Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation. J. Acoust. Soc. Am., 117(2):503–517, 2005. [2] D. Heimann. Three dimensional linearised euler model simulations of sound propagation in idealised urban situations with wind effects. Applied Acoustics, 68:217–237, 2007. [3] T. van Renterghem and D. Botteldooren. Numerical evaluation of tree canopy shape near noise barriers to improve downwind shielding. J. Acoust. Soc. Am., 123:648–657, 2007. [4] J.P. Boyd. Chebyshev and Fourier spectral methods. Dover Publications, INC., 2001. [5] B. Cott´e and Ph. Blanc-Benon. Time-domain simulations of sound propagation in a stratified atmosphere over an impedance ground. J. Acous. Soc. Am., 125(5), 2009. EL 202-207. [6] M. Hornikx, R. Waxler, and J. Forss´en. The extended Fourier pseudospectral time-domain method for atmospheric sound propagation. J. Acoust. Soc. Am., 128(4):1632–1646, 2010. [7] B. Cott´e, Ph. Blanc-Benon, C. Bogey, and F. Poisson. Time-domain impedance boundary conditions for simulations of outdoor sound propagation. AIAA J., 47(10), 2009. 2391-2403. [8] D. Gottlieb and J.S. Hesthaven. Spectral methods for hyperbolic problems. Journal of Computational and Applied Mathematics, 128:83–131, 2001. [9] L.N. Trefethen. Spectral methods in MATLAB. Society for industrial and applied mathematics, 2000. [10] J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys., 114:185–200, 1994. [11] J. Diaz and P. Joly. A time domain analysis of PML models in acoustics. Comput. Methods Appl. Mech. Engrg., 195:3820–3853, 2006. [12] F. Collino. Perfectly matched absorbing layers for the paraxial equations. J. Comp. Phys., 131:164– 180, 1997. [13] C. Bogey and C. Bailly. A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. Comp. Phys., 194:194–214, 2004. [14] D. Dragna, B. Cott´e, Ph. Blanc-Benon, and F. Poisson. Time-domain simulations of outdoor sound propagation with suitable impedance boundary conditions. AIAA J., 49(7), 2011. [15] M. Hornikx. Numerical modelling of sound propagation to closed urban courtyards. PhD thesis, Chalmers University of Technology, 2009. [16] Y. Miki. Acoustical properties of porous materials - modifications of delany-bazley models. J. Acoust. Soc. Jpn., 11(1):19–24, 1990. [17] X. Di and K. E. Gilbert. An exact Laplace transform formulation for a point source above a ground surface. J. Acous. Soc. Am. [18] E.M. Salomons, R. Blumrich, and D. Heinmann. Eulerian time-domain model for sound propagation over a finite-impedance ground surface. comparison with frequency-domain models. Acta Acustica united with Acustica, 88, 200. 483-492. [19] A. Berry and G.A Daigle. Controlled experiments of the diffraction of sound by a curved surface. J. Acous. Soc. Am., 83(6):2047–2058, 1988. [20] R.B. Platte and A. Gelb. A hybrid Fourier-Chebyshev method for partial differential equations. J. Sci. Comput., 39, 2009. 244-264.

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