PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Robust virtual sensing of the exterior noise radiation from a complex
structure in different acoustic environments
Sjoerd VAN OPHEM1, 2, a; Elke DECKERS1, 2; Wim DESMET1, 2
1

KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B, B-3001, Heverlee, Belgium
2

DMMS lab, Flanders Make, Belgium

ABSTRACT
By using a numerical model of a vibro-acoustic system in combination with a small set of measurements in a
Kalman filter, it is possible to estimate the sound pressure at locations where no microphones are present.
This can be achieved with increased accuracy as compared to using only the numerical model due to theinclusion of the expected process and measurement noise. This procedure is also known as virtual sensing. In
this paper, a model-based virtual sensor is built for a complex, deep drawn structure that radiates sound into
the free field. A finite element model is used that is extended with infinite elements to approximate theSommerfeld radiation condition. Since the original model is too large to be used in a Kalman filter, it isreduced by a stable Krylov based model order reduction technique. The performance of this virtual sensor istested in a semi-anechoic chamber, and additionally in several acoustical environments that do not conformto the Sommerfeld condition in order to validate whether the Kalman filter can handle the additional
uncertainties. The obtained results show that the filter performs well, which means that the required modeling
effort can be reduced significantly by assuming free-field conditions.
Keywords: Kalman filter, Model order reduction, Finite elements

1. INTRODUCTION
To optimize production processes and reduce downtime of machines in modern factories,
cyber-physical systems start playing a more important role (1). Cyber-physical systems arecomputational entities that interact both with each other and the physical system to which they areattached to through sensors. The exact shape of these systems can vary, depending on the available
data and knowledge of the system under investigation.
It is possible to create a physics based model when the equations of the underlying physics are
known. An accurate physics-based model allows for a detailed physical analysis of the model, thus
giving a high degree of physical insight in what is happening. Unfortunately, it is hard to construct anaccurate physics based model due to geometrical complexities, uncertainty on material properties andboundary conditions, lack of insight in what type of physics have to be modeled, etc. Thus, a highdegree of modeling expertise is required (2).
Also a purely data driven approach can be taken to construct the cyber-physical system. Thisapproach does not require any knowledge about the physics of the system and thus leads to a reduced
modeling effort. The disadvantages of this approach are that the results do not offer physical insightand that usually a large amount of data is required to train the system. Also, results are only obtainedon the locations where the sensors are placed. In many dynamic systems of interest it is desired to place
the sensors at non-intrusive locations, thus it would only be possible to measure the phenomena
happing at intrusive locations indirectly.
A third option is to utilize a hybrid approach that combines a reasonably accurate physics basedmodel with a small amount of measurements. Specifically, they are combined in a Kalman filter, which
is a state estimation algorithm that takes process noise (which represents the modeling error) andmeasurement noise (representing noise on the sensors) statistics into account to estimate the fiel dvariables in the full domain (3), thus also at locations where it might be difficult to place sensors. Since
the combination of only a small amount of measurements with the numerical modeling leads to ana

Sjoerd.vanophem@kuleuven.be4257

estimation of all the field variables, this approach is called virtual sensing.
In this paper it is chosen to investigate the effectiveness of this virtual sensing approach for exterior
vibro-acoustic problems. In the field of active noise control already several data-driven virtual sensing
approaches have been investigated (4,5). Conventional active noise control algorithms try to reduce
the sound pressure at a certain microphone location by destructive interference. However, in many
instances it is impractical to place the microphone at the location where it is desired to lower the soundpressure level. For example, at the ears of a person. Therefore, virtual sensing is used to estimate thepressure at this location, usually by means of a-priori transfer function identification. Since theseapproaches are data-driven, they only work on the pre-measured locations and do not offer flexibility.
An alternative is to use a numerical model that describes the physics. In (6) such an approach isconsidered for interior vibro-acoustic problems. They show that the combination of a Finite Element(FE) model with a small amount of measurements in a Kalman filter leads to accurate pressure
estimates in the full cavity. To reduce the calculation complexity of the FE m odel, Model Order
Reduction (MOR) is used to obtain compact models (7,8).
To be able to simulate exterior vibro-acoustic models with the Finite Element Method (FEM)
instead, the model has to adhere to the Sommerfeld radiation condition. In (9) it is explained how a
combined FE model with conjugated Astley-Leis infinite elements (10,11) on the boundary leads to amodel that approximately adheres to the Sommerfeld radiation condition and can be reduced in a stablemanner by using Krylov based MOR techniques. We utilize this technique in this paper to build avirtual sensor scheme for exterior vibro-acoustics.
The initial verification of the performance of the virtual sensor is done in an acoustic environmentthat closely represents the assumptions that are done in the numerical model (semi -anechoic
conditions). From a modelling perspective this would be a convenient assumption, even when theenvironment is not strictly anechoic, since it does not require detailed modelling of the potentiallycomplex acoustic environment, for example the modelling of the radiated sound from a structure in afactory. Therefore, the robustness of the Sommerfeld assumption on changing acoustic conditions isinvestigated by comparing the performance of the virtual sensor under correct assumptions(measurements in a semi-anechoic room) with incorrect assumptions (measurements in an off ice andin an entrance hall).

2. System formulation
The considered structure is an oil-pan, which is shown in Figure 1. The structure was produced bya deep-drawing process, leading to uncertainty on the final shape, the thickness and the material
properties (12). The oil-pan is modeled with a structural FE model consisting of quadratic shellelements, that is divided into 12 patches, as is shown in Figure 1. Each of these p atches has a separate
Young’s modulus and thickness. Before this model is coupled to an acoustic mesh , a model updatingprocedure is run to update the Young’s modulus of the patches so that the structural frequency response
functions at several locations match approximately until 500 Hz, see Figure 2. The final thicknesses
and Young’s moduli are shown in Table 1. These values do not correspond to real physical values,
since in the model updating step other uncertain parameters are not updated, such as geome tricaldeviations between the real geometry and the used numerical model.

Figure 1 - Structure under investigation (left). Division into patches (right).
An acoustic mesh finite element is added that consists of a half sphere ( ܴ ൌ ͲǤ͹ͷ m) with4258

conjugated Astley-Leis infinite elements of 10 th order attached on the surface of the sphere to model
the acoustic radiation to infinity. On the bottom surface a perfectly reflecting boundary condition isadded. Both the structural and acoustic mesh have at least 6 elements per wavelength, leading to amodel with a total of 71259 Degrees Of Freedom (DOFs). A small amount of proportional damping is
added to the structure (ߙ௦ ൌ ͳǡ ߚ௦ ൌ ͳ݁ െ ͻ) to make sure that the response as function of time decays
to zero. The resulting system of equations can be written as follows:ሷ ሺሻ ൅ ሶ ሺሻ ൅ ሺሻ ൌ ሺሻǡ‫ݕ‬ሺ‫ݐ‬ሻ ൌ ‫ݔܮ‬ሺ‫ݐ‬ሻǡ

(1)
(2)

in which ‫ܯ‬ǡ ‫ܥ‬ǡ ‫ א ܭ‬Թ௡ൈ௡ are the mass, stiffness and damping matrices respectively that follow
directly from the finite element model, ‫ܨ‬ሺ‫ݐ‬ሻ ‫ א‬Թ௡ൈଵ is the time dependent input, ‫ א ܮ‬Թ௡బൈ௡ is theoutput matrix with ݊௢ outputs and ‫ݔ‬ሺ‫ݐ‬ሻ ‫ א‬Թ௡ൈଵ is the vector of field variables to be solved for.

Figure 2 – Structural FRFs before and after model updating. The location of the excitation point is indicatedin Figure 4.
Table 1 – Structural parameters after model updating.
Patch
Thickness(mm)
Young’smodulus
(GPa)

1

2

3

4

5

6

7

8

9

10

11

12

1.7

1.5

1.7

1.4

1.5

1.5

1.5

1.4

1.3

1.3

1.3

1.4

260

78

17

230

1023

249

85

340

219

379

147

350

2.1 Model order reduction
Since the system size resulting from FE discretization is far too large to be solved in (near -)realtime, MOR techniques are used to reduce the system size. The aim of the MOR scheme is to reduce
the size to a size ݇ ‫݊ ا‬, while retaining the important dynamics. This is done by calculating a reduced
basis ܸ ‫ א‬Թ௡ൈ௞ that projects the matrices on a subspace of the system matrices as follows:୰ ൌ  ୘ ǡ  ୰ ൌ  ୘ ǡ୰ ൌ  ୘ ǡ  ୰ ൌ  ୘ ǡ୰ ൌ ǡ ൌ ୰ ǡ

(3)
(4)
(5)

which leads to the following system of reduced equations:୰ ሷ ௥ ሺሻ ൅ ୰ ሶ ௥ ሺሻ ൅  ୰ ୰ ሺሻ ൌ୰ ൌ ‫ܮ‬௥ ‫ݔ‬௥ Ǥ୰ ሺሻǡ

(6)(7)

The reduced basis can be obtained in several ways. In this paper a Krylov based MOR technique isused to calculate ܸ. This method aims to approximate the transfer function of the system around a setof expansion points by an implicit moment matching procedure. The moments are equivalent to thederivatives of the transfer function at the chosen expansion point. This method works well for large
sparse matrices, such as the matrices that result from FE models. Details about the used algorithm, the4259second order Arnoldi procedure, can be found in Refs. (8,13). To retain stability under reduction, thestrategy described in (6,9) is followed. The application of this procedure to the full order model leads
to a ROM of 42 DOFs that is accurate until 600 Hz (meaning a relative error below 0.1% for all theconsidered frequencies), thus a very significant reduction.
2.2 Conversion to first order form
To implement the derived model in a Kalman filter (see Section 3), the system has to be rewritten to
first order form. This is done by rewriting the system to an equivalent system that has twic e the amountof DOFs.
‫ݔ‬ሶ ሺ‫ݐ‬ሻ‫ ݔ‬ሺ‫ݐ‬ሻ
Ͳ
‫ܫ‬
Ͳ
(8)
൤ ௥ ൨ൌ൤൨ ൤ ௥ ൨ ൅ ൤ ିଵ ൨ ‫ݑ‬ሺ‫ݐ‬ሻǡെ‫ܯ‬௥ିଵ ‫ܭ‬௥ െ‫ܯ‬௥ିଵ ‫ܥ‬௥ ‫ݔ‬ሶ ௥ ሺ‫ݐ‬ሻ‫ܯ‬௥ ‫ܨ‬௥‫ݔ‬ሷ ௥ ሺ‫ݐ‬ሻ‫ݔ‬௥ ሺ‫ݐ‬ሻ(9)୰ ሺ‫ݐ‬ሻ ൌ ሾ‫ܮ‬௣௢௦‫ܮ‬௩௘௟௥ ሿ ൤‫ݔ‬ሶ ሺ‫ݐ‬ሻ൨Ǥ௥௥
We will rewrite this using the following shorthand notation:௖୰ ൌ ൤Ͳെ‫ܯ‬௥ିଵ ‫ܭ‬௥‫ܫ‬Ͳ൨ ǡ ‫ܤ‬௥௖ ൌ ൤ ିଵ ൨ ǡ ‫ܮ‬௖௥ ൌ ሾ‫ܮ‬௣௢௦௥െ‫ܯ‬௥ିଵ‫ܥ‬௥‫ܯ‬௥ ‫ܨ‬௥‫ܮ‬௩௘௟௥ ሿǡ

(10)in which the superscript ܿ indicates that the matrices are defined in continuous time. To discretize
these matrices several strategies can be used. In this paper it is chosen to use an exponential integration
scheme which leads to the following system of equations (for a sampling rate ݂௦ ൌ ͳȀܶ௦ ):ୢ୧ାଵൌ ‫ܣ‬ௗ ‫ݔ‬௜ௗ ൅ ‫ܤ‬ௗ ‫ݑ‬௜ௗ ǡ୧ୢ ൌ ‫ݔܮ‬௜ௗ ǡ

with

೎

(11)
(12)

షభ

(13)

ୢ ൌ ݁ ஺ೝ்ೞ ǡ ‫ܤ‬ௗ ൌ ‫ܣ‬௖௥ ሺ‫ܣ‬ௗ െ ‫ܫ‬ሻ‫ܤ‬௥௖ Ǥ

The current sample is indicated with subscript ݅. Note that the exponential in Eq. (13) is a matrix
exponential. While the matrix exponential is far too expensive to calculate for the full order model, thecalculation can be easily done for the ROM.

3. Kalman filter
The virtual sensor works by using a state estimation algorithm. This algorithm has the aim toestimate the internal states, such as pressure, acceleration, displacement, etc., by using the derived
numerical model in combination with a limited set of measurements to get an estimation that is better
than the prediction of merely the numerical model. Since the used numerical model is linear, it ischosen to use a Kalman filter as state observer. The Kalman filter explicitly takes measurement andprocess (model) noise into account as follows: They are both assumed to have a Gaussian distribution
ࣨ with a zero mean, leading to covariance matrices ܳ௜ ǣ ‫ݓ‬௜ ̱ࣨሺͲǡ ܳ௜ ሻ and ܴ௜ ǣ ‫ݒ‬௜ ̱ࣨሺͲǡ ܴ௜ ሻ. The
Kalman performs a two-step procedure. Firstly, a prediction step is done:
Predicted state estimate:
Predicted error covariance:‫ݔ‬ො ௗ ௜ȁ௜ିଵ ൌ ‫ܣ‬ௗ ‫ݔ‬ො ௗ ௜ିଵȁ௜ିଵ ൅ ‫ܤ‬ௗ ‫ݑ‬௜ ǡௗௗ்ܲ௜ȁ௜ିଵ ൌ ‫ܲ ܣ‬௜ିଵȁ௜ିଵ ‫ܣ‬൅ ܳ௜ Ǥ

(14)(15)

In this step the estimate from the previous step is used to get an estimate of the current time step.
The second step corrects this estimate by using the measurement ‫ݖ‬௜ :Innovation:
Innovation covariance:
Kalman gain:
Updated state estimate:
Updated error covariance:
Measurement post-fit residual:

4260‫ݕ‬෤௜ ൌ ‫ݖ‬௜ െ ‫ݔܮ‬ො ௗ ௜ȁ௜ିଵ ǡܵ௜ ൌ ܴ௜ ൅ ‫ܲܮ‬௜ȁ௜ିଵ ‫ ்ܮ‬ǡ‫ܭ‬௜ ൌ ܲ௜ȁ௜ିଵ ‫ܵ ்ܮ‬௜ିଵǡௗ‫ݔ‬ො ௜ȁ௜ ൌ ‫ݔ‬ො ௗ ௜ȁ௜ିଵ ൅ ‫ܭ‬௜ ‫ݕ‬෤௜ ǡܲ௜ȁ௜ ൌ ሺ‫ ܫ‬െ ‫ܭ‬௜ ‫ܮ‬ሻܲ௜ȁ௜ିଵ ǡ‫ݕ‬෤௜ȁ௜ ൌ ‫ݖ‬௜ െ ‫ݔܮ‬ො ௗ ௜ȁ௜ Ǥ

(16)
(17)
(18)
(19)
(20)
(21)

To make the Kalman filter stable, the system has to be fully detectable (14). Detectability of thesystem can be guaranteed by choosing the right sensors. Furthermore, the sensor locations for the best
virtual sensor performance can be calculated by using this metric, as has been explained in detail in
Refs. (15–17).

4. Virtual sensing performance
4.1

Performance under ideal acoustic conditions

Figure 3 - Oil-pan setup in semi-anechoic room.
Before the performance of the filter under non-ideal conditions is assessed, firstly the performanceof the filter is evaluated under ideal conditions. The oil-pan was placed in a semi-anechoic room at theKU Leuven Sound and Vibration Lab, see Figure 3. Thus, the setting of the real test setup has the same
boundary conditions as the numerical model described in Section 2. Six microphones were placedaround the structure and act as measurements that are used by the Kalman filter (their locations appear
in matrix ‫)ܮ‬. Furthermore, 4 microphones are used as reference microphones. These microphones are
not used in the filter, but are only used to assess the estimation performance of the virtual sensor. The
microphone locations are indicated in Figure 4.

Figure 4 – Microphone and excitation locations (indicated with A1, A2). The microphones that are used in
the Kalman filter are indicated in a bolt font. The other microphones only function as verification of theresults.
The structure is excited with a modal hammer and both the excitation force and acoustic response isused in the Kalman filter. A sample rate of ݂௦ ൌ ͶͲͻ͸ Hz is used throughout the experiment and alow-pass filter is applied on the signal with a cut-off frequency at 500 Hz, since in that frequency range
the structural model gives reasonably accurate results. The covariance of the model uncertainty is set
to 3e-6 and the covariance of the measurement uncertainty is set to 4.2e -2, since these values lead to4261

the best performance. The results are shown in Figure 5, where the measured acoustic pressure signals
at the reference microphones are compared to the estimated pressure signals. In the figure only
microphone 6 and and 8 are shown due to space constraints in this paper, but a more in -depth analysis
of a similar analysis can be found in (16). It can be observed that the virtual sensor performance issignificantly better than the prediction from the numerical model alone, thus the virtual sensor can
successfully estimate the pressure signal at locations where no microphone is placed. Although notdetailed in this paper, the same virtual sensing scheme could also be used for the estimation of otheracoustic quantities, such as the particle velocity, the acoustic intensity and the acoustic power. Details
can be found in Refs. (16,17).

Figure 5 – Virtual sensor estimation under ideal conditions.
4.2 Performance under non-ideal acoustic conditions
To assess the robustness of the above procedure for other acoustic environments, which are only
approximately behaving as a semi-infinite free field, the above experiment with the same sensor
placement is repeated in two different acoustic environments, namely an office (around 3x6x3 m(length, width, height)) and an entrance hall (around 10x20x10 m (length, width, height)) of auniversity building. The setup at these locations is shown in Figure 6.

Figure 6 – Oil pan setup in office (left) and entrance hall (right).
Both of the setups are placed on a table, instead of on the floor, which is modelled as a perfectlyreflecting surface. The main assumption that is done here is that the Kalman filter is able to capture theuncertainties in the acoustic domain, due to the noise assumptions of the filter. The possibility of using
the Kalman filter for these non-ideal environments would increase the potential applications for thesefilter, since it would allow for the estimation of the acoustic pressure in many situations (for example
structures in large halls) that would otherwise need a complex acoustic model. The results are shown in
Figure 7.

4262

Figure 7 – Virtual sensing performance under non-ideal conditions.

It can be observed that the virtual sensor still manages to capture the global dynamic behavior,although the accuracy of the estimation is less than for the ideal case, which is e xpected.

5. Conclusions
In this paper it is shown how an exterior vibro-acoustic virtual sensor can be created by combining
a numerical model with a small amount of microphone measurements. The numerical model consistsof finite elements, combined with infinite elements to approximate the Sommerfeld radiation
condition. To allow for (near-)real time virtual sensing, the numerical model is reduced in size withstate-of-the-art model order reduction techniques. The measurements and the reduced numericalmodel are combined in a Kalman filter for acoustic pressure estimation in the full domain.
The effectiveness of the proposed virtual sensor is shown with several experiments. Firstly, thegeometry under investigation, which is an oil-pan-like structure, was placed in a semi-anechoic room.
This is considered to be the best case scenario, since the acoustic boundary conditions of the numericalmodel correspond to the experimental setup. It is shown that the virtual sensor gives goodcorrespondence between the estimated pressure and the measured pressure.
Furthermore, the same setup was placed in two non-ideal acoustic scenarios that do not behave likesemi-infinite free spaces, while the numerical model was kept the same, thus with the Sommerfeld
assumption, to see if this additional model uncertainty can be captured with the noise assumption of
the Kalman filter. It is shown that the virtual sensing performance for both scenarios is still acceptable,
although the estimation quality is less than for the ideal case. Further research will be devoted to amore in-depth investigation of how this uncertainty can be captured, for example by utilizing a Kalmanfilter with colored process noise, instead of white noise, so that the performance of the filter can be
improved when it is used in a non-ideal acoustic environment.

ACKNOWLEDGEMENTS
The Research Fund KU Leuven is gratefully acknowledged for its support. The research of E.
Deckers is funded by a grant from the Research Foundation - Flanders (FWO).

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