Document [original]

This version is available at https://doi.org/10.14279/depositonce-8684

right to use is granted. This document is intended solely for

personal, non-commercial use.

Lemke, M.; Straube, F.; Stein, L.; Weinzierl, S. (2018): Optimized Sound Field Generation in the Time

Domain - Validation for Source Arrays in 2D. In: Fortschritte der Akustik - DAGA 2019 : 45. Jahrestagung

für Akustik, 18.-21. März 2019 in Rostock. Berlin: Deutsche Gesellschaft für Akustik e.V., 2018. pp.

1442–1445.

Mathias Lemke, Florian Straube, Lewin Stein, Stefan Weinzierl

Optimized Sound Field Generation in the

Time Domain - Validation for Source

Arrays in 2D

Published versionConference paper |

Optimized Sound Field Generation in the Time Domain –

Validation for Source Arrays in 2D

Mathias Lemke1, Florian Straube2, Lewin Stein1, Stefan Weinzierl2

1Institut f¨ur Str¨omungsmechanik und Technische Akustik, TU Berlin, Germany, Email: [email protected]erlin.de

2Fachgebiet Audiokommunikation, TU Berlin, Germany

Introduction

Line Source Arrays (LSAs) are used for sound reinforce-

ment aiming at the synthesis of homogeneous sound

fields for the whole audio band width. The deployed

loudspeaker cabinets are rigged with different tilt an-

gles and/or are electronically controlled. The determina-

tion of the optimal geometric arrangement and electronic

drive is an ill-posed inverse problem.

In two preceding contributions [5, 6], we introduced an

adjoint-based approach for sound reinforcement applica-

tions in the time domain. By defining a target sound

field within an objective function the method allows the

optimization of acoustic sources also considering loud-

speaker directivities, a base flow or thermal stratification.

It is based on the Euler equations and the correspond-

ing adjoint which are solved by means of computational

aeroacoustic (CAA) techniques. Both, appropriate driv-

ing functions and appropriate positions of the sources for

the generation of a desired sound field can be determined.

In the following, we will present new validation examples

for two-dimensional arrangements: a circular array con-

sisting of monopole loudspeakers, and one speaker with

a circular piston directivity. It will be shown that pre-

specified driving functions – amplitudes and phases – can

be regained or that the corresponding target sound fields

can be determined using the adjoint-based method.

Governing equations

The governing equations for the computations are the

two-dimensional Euler-equations [3], abbreviated by E

in terms of a space-time-solution. The full state, in-

cluding density, velocities and pressure, as energy quan-

tity, is accumulated in the state vector q. In order to

model acoustic loudspeakers, the equation is extended

by a right-hand-side f(xi, t), only affecting the pressure

equation.

E(q) = f(1)

The all-encompassing idea is to optimize this term by

means of an adjoint-based approach and, thus, to de-

termine optimal driving functions for real loudspeaker

arrays.

Adjoint-based approach

The required high-dimensional gradient for optimization

of the loudspeakers – represented by fi(t) – is determined

by a continuous adjoint-based approach, cf. [3], which

aims at minimizing the following objective function, (2)

and (3). The objective is defined as integral difference

between a current solution of the Euler-equations qand

a desired target state qtarget.

J=1

2ZZ

Ω

(q−qtarget)2dΩ (2)

δJ =ZZ

Ω

(q−qtarget)

| {z }

δqdΩ = gTδq (3)

By combining the (linearized) Euler-equations Elin and

the (linearized) objective in a Lagrangian manner, and

a simple rearrangement the adjoint Euler-equations are

defined.

δJ =gTδq −(q∗)T(Elinδq −δf)

| {z }

(4)

=δqT(g−ET

linq∗)

| {z }

=0,adjoint eqn.

+(q∗)Tδf (5)

Thus, the change of the objective becomes independent of

the change of the state δq. The required high-dimensional

gradient results in

δJ =q∗Tδf →δJ

δf =q∗≈ ∇fJ. (6)

It is given by the adjoint solution. More details on the

derivation of the adjoint Euler-equations can be found

in [3, 4, 5]

The adjoint-based gradient is used in an iterative man-

ner, see Fig. 1. The desired optimized driving functions

for the speakers are obtained by deconvolution of the op-

timal fiwith the input signal creating the target sound

field.

Computational setup

The governing and the corresponding adjoint equations

are discretized by finite differences. A sixth-order accu-

rate compact symmetric derivation stencil is used. The

sound reinforcement area under consideration ranges over

1.2 m ×1.2 m and is discretized by a uniform grid which

consists of 181 ×181 grid points. The computational

time span is separated into equidistant time steps using

a CFL-condition equal to 0.96, which corresponds to a

sampling rate of ∼53.33 kHz. An explicit fourth-order

Runge-Kutta scheme is used for the time-wise integra-

tion. All boundaries are treated as non-reflecting using

characteristic boundary conditions in combination with a

quadratic sponge layer. Initial and boundary conditions

are chosen to realize a speed of sound equal to 343 m/s.

DAGA 2019 Rostock

1442

initial guess

f 0=0

direct

solution

N(q, f n)

target

qtarget

adjoint

solution

N*(q,q*,Δ q)

gradient

update f n+1

- line search

- CG methods

Δ q = q - qtarget

optimal

convergence

optimal

drive

deconvolution

optimal

transformation

Figure 1: Iterative process for determining optimal driving functions: Starting from an initial guess for the loudspeakers, the

governing equations are solved forward in time. The result is compared to the desired target state, while the actual difference

drives the adjoint equations, which are solved backward in time. Based on the adjoint solution, the gradient ∇fJis used to

update the actual forcing. Once convergence is reached, deconvolution of optimal f’s with the target input signal results in

optimal drives for the speakers.

Figure 2: Setups under consideration: (Left): 12 monopole speakers in circular arrangement. (Right): Single speaker with

circular piston directivity. In both graphics, the dashed contour lines mark the area, where the objective function is evaluated.

Figure 3: (Left): Amplitude frequency responses of the reference input signals. (Right): Corresponding curves resulting from

adjoint-based optimization. Please note the different definitions: resulting pressure fluctuation (Left) and forcing amplitude

(Right).

Results

Two different configurations are examined. The first con-

figuration includes 12 monopole speakers arranged in a

circular manner, see Fig. 2. The objective is restricted

to the area enclosed by the speakers. As reference input

signal, a logarithmic sweep from 1 kHz to 3 kHz is used.

After 30 iterations the reference sound field, encoded

DAGA 2019 Rostock

1443

Figure 4: (Left): Amplitude frequency response of speaker (1) using quadratic conversion for the optimization results. Within

the considered frequency range from 1 kHz to 3 kHz, the deviation is smaller than 1 dB. (Right): Phase frequency response of

speaker (1) using a constant phase shift of π/2. Within the considered frequency range from 1 kHz to 3 kHz, the deviation is

smaller than 0.15 rad.

Figure 5: (Left): The reference sound field and (Right): the optimized sound field after 30 iterations of the adjoint-based

approach for the time steps 1500 (Top) and 2250 (Bottom) for the circular array configuration. Please note, the sponge region

is also shown.

in qtarget, is reproduced, see Fig. 5. The frequency-

dependent amplitude responses for all speakers are iden-

tified, see Fig. 3. Within the sweep frequency range,

the optimized amplitude ratios are in accordance with

the reference ratios. The required conversion from dis-

turbance (in the reference curve) to source definition (as

result of the optimization) is identified as quadratic func-

tion corresponding to the analytic solution [1].

A detailed analysis of a single speaker, marked in red in

Fig. 2, proves that the optimized amplitude and phase

responses vastly match the reference ones, see Fig. 4.

The resulting amplitude deviations are smaller than 1 dB

within the sweep frequency range. The results are repre-

sentative for all speakers.

The second selected configuration consists of a single

speaker centered in the sound reinforcement area. This

speaker features a circular piston directivity. In or-

der to reproduce the resulting target sound field, the

DAGA 2019 Rostock

1444

Figure 6: (Left): The reference sound field and (Right): the optimized sound field after 150 iterations of the adjoint-based

approach for the time steps 800 (Top) and 1150 (Bottom) for the single speaker configuration with circular piston directivity.

Please note, the sponge region is also shown.

adjoint-based framework is used to optimize more than

60 monopole sources, located at the grid nodes in the

center region of the computational domain, following the

monopole synthesis approach [2, 7].

In Fig. 6, it can be seen, that the reference sound field

is reproduced in good quality. Using more monopole

sources for the synthesis improves the results, while re-

quiring the same computational time. Thus, it is shown,

that the adjoint-based approach is capable of handling

loudspeaker directivities.

References

[1] K. Ehrenfried. Str¨omungsakustik: Skript zur Vor-

lesung. Berliner Hochschulskripte. Mensch-und-Buch-

Verlag, 2004.

[2] J. Escolano and J. J. Lopez. Directive sources

in acoustic discrete-time domain simulations based

on directivity diagrams. J. Acoust. Soc. Am.,

121(6):EL256–EL262, 2007.

[3] M. Lemke. Adjoint based data assimilation in com-

pressible flows with application to pressure determi-

nation from PIV data. PhD thesis, Technische Uni-

versit¨at Berlin, 2015.

[4] M. Lemke, J. Reiss, and J. Sesterhenn. Adjoint based

optimisation of reactive compressible flows. Combus-

tion and Flame, 161(10):2552 – 2564, 2014.

[5] M. Lemke, F. Straube, F. Schultz, J. Sesterhenn, and

S. Weinzierl. Adjoint-based time domain sound rein-

forcement. In Audio Engineering Society Conference:

2017 AES International Conference on Sound Rein-

forcement – Open Air Venues, Aug 2017.

[6] M. Lemke, F. Straube, J. Sesterhenn, and

S. Weinzierl. Adjungierten-basierte Schallfeldsyn-

these und Beschallung. In Fortschritte der Akustik -

DAGA 2017, pages 1422–1425. Deutsche Gesellschaft

f¨ur Akustik e.V., 2017. ISBN: 978-3-939296-12-6.

[7] M. Ochmann. The source simulation technique for

acoustic radiation problems. Acustica, 81:512–527,

1995.

The authors gratefully acknowledge the support of the

Deutsche Forschungsgemeinschaft (DFG) within the

projects LE 3888/2-1 and WE 4057/16-1.

DAGA 2019 Rostock

1445