Highly accurate predictions from large-scale numerical simulations are associated with increased computational resources and time expense. Consequently, the data generation process can only be performed for a small sample size, limiting a detailed investigation of the underlying system. The concept of multi-fidelity modeling allows the combination of data from different models of varying costs and complexities. This study introduces a multi-fidelity model for the acoustic design of a vehicle cabin. Therefore, two models with different fidelity levels are used to solve the Helmholtz equation at specified frequencies with the boundary element method. Gaussian processes (GPs) are trained on each fidelity level with the simulation results to predict the unknown system response. In this way, the multi-fidelity model enables an efficient approximation of the frequency sweep for acoustics in the frequency domain. Additionally, the proposed method inherently considers uncertainties due to the data generation process. To demonstrate the effectiveness of our framework, the multifrequency solution is validated with the high-fidelity (HF) solution at each frequency. The results show that the frequency sweep is efficiently approximated by using only a limited number of HF simulations. Thus, these findings indicate that multi-fidelity GPs can be adopted for fast and, simultaneously, accurate predictions.

A key aspect in acoustic design processes is the fast and accurate characterization of the acoustic system. Rising requirements on modern products demand a drastic increase in the complexity of the model. Consequently, design tasks, such as sensitivity analyses and structural-acoustic optimizations, are generally associated with high computational costs (Marburg, 2002). Thus, engineers and researchers are faced with the challenge of finding solutions for large problems on the basis of small data. Ideally, their models should be fast to evaluate and, at the same time, highly accurate. However, models with a high predictive quality are generally accompanied with high demands on resources and time, whereas models, which are fast to evaluate, are usually less accurate.

Traditionally, detailed analyses are performed to assess the acoustic properties of a system. These can either be cumbersome physical experiments, e.g., measurements in an impedance tube or a reverberation chamber, or expensive numerical simulations with the finite element method (FEM) or boundary element method (BEM). At the same time, various analytical and numerical models at small scale exist, which can already yield a decent approximation of the relevant acoustic solution. One specific problem in acoustics is associated with the approximation of the frequency sweep when using BEM simulations. Existing methods, e.g., greedy approximation (Baydoun , 2020; Baydoun , 2021; Jelich , 2021) or parametric model order reduction (Panagiotopoulos , 2022; Xie and Liu, 2021; Xie , 2022; Xie , 2023), already provide an efficient multifrequency solution technique. However, uncertainties, whether in the model parameters or due to the random nature of the data generation process, are not quantified within these methods. The efficient solution for a wide frequency range under uncertainties is particularly important in the acoustic design of a vehicle cabin (Gurbuz , 2022a; Schmid , 2022). Moreover, recent evidence suggests to prefer the sound energy density to the sound pressure as a control objective in interior acoustic problems (Cazzolato and Hansen, 1998; Gurbuz and Marburg, 2022; Gurbuz , 2022b; Koopmann and Fahnline, 1997; Sommerfeldt and Nashif, 1994; Tanaka and Kobayashi, 2006). With the sound energy density being sensitive to the sound pressure and particle velocity, the sound energy density provides a robust objective function.

Gaussian processes (GPs) provide a useful technique to substitute a complex model by an efficient surrogate model (Hoffer , 2022). In the sense of a Bayesian method, GPs allow to embed prior knowledge of the underlying problem and further enable predictions under consideration of uncertainties (Cutajar , 2019; Williams and Rasmussen, 2006). In particular, GPs account for two types of uncertainties: aleatoric and epistemic. Aleatoric or statistical uncertainties refer to the random nature of the data generation process. For instance, the output of an experiment changes when repeated. In contrast to this, epistemic or systematic uncertainties are defined as uncertainties resulting from limited knowledge of the underlying problem. In this case, it involves simplifying assumptions on the model or limited information on the model parameters. As a matter of fact, epistemic uncertainties can be mitigated by incorporating additional information. Aleatoric uncertainties, on the other hand, would remain unaffected here (Hüllermeier and Waegeman, 2021; Soize, 2017).

Raissi (2017b) reported on GPs which are used to solve linear partial differential equations. These authors further extended their method to time-dependent and nonlinear problems (Raissi , 2018). In the field of acoustics, GPs were adopted as a surrogate model in a transfer path analysis of a vehicle cabin (Gurbuz , 2022a). GPs were further used as an alternative to Bayesian inference (Schmid , 2021) to infer a two-dimensional sound field based on a limited number of sound pressure observations (Caviedes-Nozal , 2021). In ocean acoustics, GPs have been applied in a source localization technique (Michalopoulou , 2021) and the quantification of uncertainties (Michalopoulou and Gerstoft, 2022; Yardim , 2011). Further source localization methods based on GPs have been reported for the sound emitted by a complex aerospace component (Hensman , 2010) and general two-dimensional sound radiation problems (Albert and Rath, 2020). In the latter work, the authors implemented a physics-informed approach. In a more recent study, GPs were integrated in a framework for the detection and characterization of cracks in rocks (Jiang , 2022).

A multi-fidelity model allows the combination of multiple models with differing fidelity levels. Typically, it consists of a low-fidelity (LF) and high-fidelity (HF) model. LF models can be attributed to low computational costs and decreased accuracy, whereas HF models can achieve predictions with higher accuracy for the burden of high expenses. In many engineering fields, analytical or numerical models at small scale can be regarded as LF models. On the other hand, highly resolved numerical models or cumbersome physical experiments can be considered to be HF models. As such, the advantages of both fidelity levels, namely, fast evaluations and high accuracy, are merged in a multi-fidelity model.

The first serious discussion and analysis of multi-fidelity modeling emerged in 1998 with the work of Craig (1998), introducing multi-fidelity models on the basis of linear regression. This model was further extended by using Bayesian linear regression for high-dimensional problems (Cumming and Goldstein, 2009). Another landmark study has been proposed by Kennedy and O'Hagan (2000), who developed a multi-fidelity model based on GPs. This method was further improved by a recursive formulation for the different fidelity levels (Le Gratiet and Garnier, 2014). Marburg and Hardtke (2001) applied the idea of multiple models with different predictive capabilities on an acoustic design task, yet, without using the term multi-fidelity. More recent attention has focused on multi-fidelity models for the solution of partial differential equations (Parussini , 2017; Raissi , 2017a). Beyond GPs, artificial neural networks have been implemented in multi-fidelity schemes for parameter-dependent outputs (Guo , 2022) or discontinuities between the fidelity levels (Raissi and Karniadakis, 2016). This type of multi-fidelity model was further applied to a structural health monitoring problem (Torzoni , 2023).

The aim of this study is the development of an efficient multifrequency solution strategy for accurate predictions under the consideration of uncertainties. For this purpose, a multi-fidelity model comprising two levels of fidelity is deployed. Regarding the LF and HF models, GPs are adopted as surrogates. These surrogates are trained on the sound field data obtained by the BEM. Therefore, the acoustic Helmholtz equation is solved for two different refinements of the boundary element mesh. The solution of the fine mesh is considered to be the HF data while the frequency response of the coarse mesh is adopted as the LF data. To validate the proposed multi-fidelity model, a vehicle interior noise problem is investigated. As recently suggested, we analyze two objective functions: the sound pressure and sound energy density (Gurbuz , 2022b). By using a multi-fidelity model, a high predictive quality can be ensured for only a small number of BEM simulations.

This paper is structured as follows. Section II briefly outlines the Helmholtz equation, BEM, and relevant acoustic field quantities, i.e., the sound pressure and sound energy density. Section III provides the fundamentals of GPs. In Sec. IV, the theoretical background of multi-fidelity models is provided. The results for an industrial vehicle interior noise problem are presented in Sec. V. The conclusions are drawn in Sec. VI.

The acoustic problem setup and BEM are only briefly introduced here. For a detailed description, the interested reader is referred to Marburg (2018), Marburg (2008), and Wu (2002). By assuming the harmonic time dependence, e i ω t, the interior acoustic problem in the frequency domain is governed by the Helmholtz equation,
Δ p ( x ) + k 2 p ( x ) = 0 , x Ω 3 ,
(1)
with the scalar, complex-valued sound pressure, p ( x ), in the domain, Ω. The wavenumber is denoted by k = ω / c with the angular frequency, ω, and speed of sound, c. In this study, we focus on air-filled domains. Thus, we introduce the index “a” to refer to the material properties of air, i.e., for the speed of sound, c = ca, and density, ρ = ρ a.
The Robin boundary condition is formulated as (Suzuki , 1989)
v f ( x ) v s ( x ) = Y ( x ) p ( x ) , x Γ 2 ,
(2)
where v f ( x ) , v s ( x ), and Y ( x ) denote the normal fluid particle velocity, normal structural velocity, and boundary admittance at the boundary, Γ. The sound pressure on the boundary is related to the normal fluid particle velocity by the linearized Euler equation,
v f ( x ) = 1 i ω ρ a p ( x ) n ( x ) , x Γ 2 ,
(3)
where the ambient density of the acoustic medium is ρa. The normal structural particle velocity is used to impose a vibrating pattern on the surface, whereas the Robin boundary condition is adopted to model the absorbing behavior of the enveloping structure.
The Kirchhoff-Helmholtz boundary integral equation reads
c ( y ) p ( y ) + Γ G ( x , y ) n ( x ) p ( x ) d Γ ( x ) = i ω ρ a Γ G ( x , y ) v f ( x ) d Γ ( x ) , x , y Γ 2 ,
(4)
where the Green's function is G ( x , y ) = e i k r / ( 4 π r ) and the Euclidean distance is r = r ( x , y ). The quantity, c ( y ), represents a geometric entity, which is determined by the boundary contour at the position, y . On smooth boundaries, c ( y ) is equal to 0.5 (Marburg, 2018). Finally, substituting Eq. (2) into Eq. (4) and discretizing the resulting equation with the collocation method yields the following linear system of equations (Marburg, 2018; Marburg, 2008):
[ H ( k ) G ( k ) Y ( k ) ] p ( k ) = G ( k ) v s ( k ) .
(5)
The entity, G ( k ), denotes the matrix of the single layer potential, whereas H ( k ) is referred to as the matrix of the double layer potential. These matrices are neither Hermitian nor positive definite. As the entries of the system matrices, G ( k ) and H ( k ), are determined by the Green's function and its normal derivative, respectively, it becomes obvious that the system matrices depend on the wavenumber and, thus, on the frequency. As a consequence, the sound pressure solution, p ( k ), in Eq. (5) depends on the frequency. To obtain the system response in a certain frequency range, Eq. (5) has to be solved for each discrete frequency within the relevant frequency range.
In interior acoustic problems, it is widely adopted to use the sound pressure at a specific location as an objective function to characterize the acoustic system. The sound pressure inside of a cavity can be obtained by evaluating Eq. (4) in the acoustic field, Ω,
p i ( y ) = i ω ρ a Γ G ( x , y ) v s ( x ) d Γ ( x ) Γ [ G ( x , y ) n ( x ) i ω ρ G ( x , y ) Y ( x ) ] p ( x ) d Γ ( x ) , y Ω ,
(6)
where p i ( y ) denotes the field point sound pressure. Similarly, by exploiting Eq. (3), the particle velocity at a field point is computed by
v i ( y ) = Γ G ( x , y ) v s ( x ) d Γ ( x ) [ Γ 1 i ω ρ a G ( x , y ) n ( x ) G ( x , y ) Y ( x ) ] p ( x ) d Γ ( x ) , y Ω ,
(7)
where the field point particle velocity is v i ( y ) (Wu and Seybert, 1991).

Recent evidence (Cazzolato and Hansen, 1998; Gurbuz , 2022b; Koopmann and Fahnline, 1997; Sommerfeldt and Nashif, 1994), however, suggests to use the sound energy density as a control objective because the sound pressure evaluation shows a poor performance in regions with low sound pressure values. This results from the fact that the sound pressure heavily depends on the evaluated position. Thus, a deteriorated performance is to be expected when the sound pressure is observed in the vicinity of a pressure node. As opposed to that, the sound energy density is relatively robust to the position of the evaluation point. Therefore, it was suggested as a control objective for interior acoustic problems.

With two energy forms transported in sound waves (Kinsler , 2000), the sound energy density is composed by the potential energy density,
e p ( y ) = 1 2 ρ a c a 2 p i * ( y ) p i ( y ) ,
(8)
and the kinetic energy density,
e k ( y ) = ρ a 4 v i * ( y ) v i ( y ) .
(9)
The total sound energy density is then obtained by
e t ( y ) = e p ( y ) + e k ( y ) = 1 2 ρ a c a 2 p i * ( y ) p i ( y ) + ρ a 4 v i * ( y ) v i ( y ) .
(10)
Equation (10) highlights that the sound energy density is sensitive to the sound pressure and particle velocity. In this way, the sound energy density provides a robust objective function, particularly, in regions with low sound pressure levels (SPLs). Further details on that topic are provided in Gurbuz and Marburg (2022), Gurbuz (2022b), and Preuss (2022).
A GP is defined as a collection of random variables which are jointly Gaussian distributed. As such, a GP can be interpreted as a probability distribution over functions. With h denoting the unknown frequency response function, the GP can be expressed by
h ( f ) G P ( m ( f ) , cov ( f , f ) ) .
(11)
A GP is specified by a mean function, m ( f ), and a covariance function or kernel, cov ( f , f ), where f and f denote two sets of input frequency points (Raissi , 2017b).
In this study, the squared exponential covariance function,
cov ( f , f ) = σ f 2 exp ( 1 2 l 2 ( f f ) T ( f f ) ) ,
(12)
with the characteristic length, l, and signal variance, σ f 2, is adopted. These parameters essentially specify the characteristics of the GP. They are, thus, commonly referred to as hyperparameters. Note that this covariance function depends only on the frequency input points. The covariance of the outputs, namely, the frequency response function, is only determined by the input covariance. Closer inspection at Eq. (12) reveals that the output increases when the distance between the input points decreases (Williams and Rasmussen, 2006).
The GP prior, which defines the GP without any observations, can be then expressed by
h N ( 0 , cov ( f , f ) ) ,
(13)
where N denotes a multivariate Gaussian probability distribution. Note that the observed quantities are the real-valued SPL and real-valued sound energy density level. Therefore, the real-valued multivariate Gaussian distribution, N, was chosen here. In this initial study, we assume that the frequency response function is zero prior to the involvement of any observed data. As a consequence, the mean function can be set to zero.
So far, the GP prior in Eq. (13) does not account for any physical information about the system. The great advantage of GPs emerges by incorporating observed data of the underlying physical process. Therefore, the joint probability distribution of the observed function outputs, h, and the unknown frequency responses, h *, at new frequencies, f *, are introduced as
[ h h * ] N ( 0 , [ K ( f , f ) K ( f , f * ) K ( f * , f ) K ( f * , f * ) ] ) .
(14)
The matrix, K, denotes the evaluated covariance function for the frequencies, f, at which observations are available, and frequencies, f *, at which the frequency responses are unknown. For n known frequency points, the dimension of K ( f , f ) amounts to n × n. Analogously, for n *, frequency points with unknown responses, the remaining sub-matrices, K ( f * , f ) , K ( f , f * ), and K ( f * , f * ) are of sizes n * × n , n × n *, and n * × n *, respectively. Now, by conditioning the joint GP prior on the observed data, the joint posterior probability distribution for the unknown frequency response function is expressed by
P ( h * | f * , f , h ) N ( K ( f * , f ) K ( f , f ) 1 h , K ( f * , f * ) K ( f * , f ) K ( f , f ) 1 K ( f , f * ) ) .
(15)
The interested reader is referred to the  Appendix for an outline of the conditioning process.
Equation (15) serves as a viable implementation for noiseless data. In reality, however, data generation processes are subject to noise resulting in noisy observations. In that case, the observations can be modeled with y = h ( f ) + ϵ under the assumption of additive Gaussian distributed noise. For noisy observations, the covariance function in Eq. (12) adapts to
cov ( f , f ) = K ( f , f ) + σ n 2 I = σ f 2 exp ( 1 2 l 2 ( f f ) T ( f f ) ) + σ n 2 I ,
(16)
where the noise level is σ n 2 and the identity matrix is I, i.e., ϵ N ( 0 , σ n 2 I ). The joint probability distribution of the noisy observations, y, and frequency responses, h *, of interest then reads
[ y h * ] N ( 0 , [ K ( f , f ) + σ n 2 I K ( f , f * ) K ( f * , f ) K ( f * , f * ) ] ) .
(17)
Conditioning the joint distribution in Eq. (17) on the observed data yields then the predictive posterior GP,
P ( h * | f * , f , y ) N ( h ¯ * , cov ( h * ) ) ,
(18)
where
h ¯ * = K ( f * , f ) [ K ( f , f ) + σ n 2 I ] 1 y ,
(19)
cov ( h * ) = K ( f * , f * ) K ( f * , f ) [ K ( f , f ) + σ n 2 I ] 1 K ( f , f * ) T .
(20)
The posterior GP is determined by the predictive mean, h ¯ *, and predictive covariance, cov ( h * ). By this means, the posterior GP is fully specified and can be used as a surrogate model for a complex acoustic system (Williams and Rasmussen, 2006).
To obtain the optimal set of hyperparameters, the negative log likelihood function (Williams and Rasmussen, 2006),
log P ( y | f ) = 1 2 y T ( K ( f , f ) + σ n 2 I ) ) y 1 2 log | K ( f , f ) + σ n 2 I | n 2 log 2 π ,
(21)
is minimized with respect to the characteristic length, l, signal variance, σ f 2, and noise level, σ n 2. Equation (21) estimates how likely the observation is for the given training inputs. In the literature, the term “training” is generally understood as the treatment of the optimization problem in Eq. (21).
Following Kennedy and O'Hagan (2000), we assume that the prediction of the HF model, h H ( f ), can be approximated by the solution of a lower fidelity version, h L ( f ). This property defines an autoregressive model, which is expressed by (Kennedy and O'Hagan, 2000; Le Gratiet and Garnier, 2014)
h H ( f ) = η h L ( f ) + δ ( f ) ,
(22)
with the coupling parameter, η, and two independent GPs,
h L G P ( 0 , cov L ( f , f ) ) ,
(23)
δ G P ( 0 , cov H ( f , f ) ) .
(24)
In Eqs. (22)–(24), the subscripts, L and H, denote the affiliation to the LF and HF models, respectively. For η = 0, it becomes evident that the LF and HF models are uncorrelated. In this case, the multi-fidelity model will not lead to any improvement (Raissi and Karniadakis, 2016). Note that the multi-fidelity model in Eqs. (22)–(24) is adjustable by two covariance functions, cov L and cov H. However, in this initial study, the squared exponential covariance function in Eq. (12) is adopted for both kernel functions.
By introducing the frequency and frequency response function data pairs for the LF level, [ f L , h L ], and HF level, [ f H , h H ], the multi-fidelity GP prior can be formulated as
[ h L h H ] G P ( [ 0 0 ] , [ cov L ( f L , f L ) η cov L ( f L , f H ) η cov L ( f L , f H ) η 2 cov L ( f L , f H ) + cov H ( f L , f H ) ] ) ,
(25)
or in compact form, where cov L L = cov L , cov L H = cov H L = η cov L, and cov H H = η 2 cov L + cov H,
[ h L h H ] G P ( [ 0 0 ] , [ cov L L ( f L , f L ) cov L H ( f L , f H ) cov H L ( f L , f H ) cov H H ( f L , f H ) ] ) .
(26)
By including the frequencies, f *, at which we are interested in the response function, h *, the multi-fidelity GP adapts to
[ h * h L h H ] G P ( [ 0 0 0 ] , [ cov * * ( f * , f * ) cov * L ( f * , f L ) cov * H ( f * , f H ) cov L * ( f L , f * ) cov L L ( f L , f L ) cov L H ( f L , f H ) cov H * ( f H , f * ) cov H L ( f L , f H ) cov H H ( f H , f H ) ] ) .
(27)
Conditioning Eq. (27) on the observed LF and HF data then yields the predictive posterior GP for the unknown frequency responses,
P ( h * | f * , f L , h L , f H , h H ) = N ( K * K 1 h , K * * K * K 1 K * T ) ,
(28)
with
h = [ h L h H ] T ,
(29)
K * = [ cov * L ( f * , f L ) cov * H ( f * , f H ) ] ,
(30)
K = [ cov L L ( f L , f L ) cov L H ( f L , f H ) cov H L ( f H , f L ) cov H H ( f H , f H ) ] ,
(31)
K * * = [ cov * * ( f * , f * ) ] .
(32)
Equations (28)–(32) fully describe the posterior multi-fidelity GP for noise-free predictions at the frequencies, f *, where we are interested in the frequency responses.
For noisy observations, y L and y H, the predictive posterior distribution is expressed by
P ( h * | f * , f L , y L , f H , f H ) = N ( K * K 1 y , K * * K * K 1 K * T ) ,
(33)
where y = [ y L y H ]. In addition to this, the covariance matrix regarding the training points adapts to
K = [ cov L L ( f L , f L ) + σ n L 2 I cov L H ( f L , f H ) cov H L ( f H , f L ) cov H H ( f H , f H ) + σ n H 2 I ] ,
(34)
with the noise levels, σ n L 2 and σ n H 2, inherent to the LF and HF data, respectively.
Regarding the magnitudes of the acoustic quantities, the noise level transfers to the signal-to-noise ratio (SNR) according to
SNR = 10 log ( y y T σ n 2 ) .
(35)
In the first study, the SPL at the driver's ear is adopted as the objective function. In the second case, the sound energy density level at the same field point is considered.
Last, the negative log likelihood function is rewritten as
log P ( y | f ) = 1 2 y T [ K ( f , f ) + σ n 2 I ] 1 y 1 2 log | K ( f , f ) + σ n 2 I | n L + n H 2 log 2 π
(36)
for the multi-fidelity GP with noisy observations, y. Analogously, the input frequency points and noise levels for the LF and HF data are comprised in f = [ f L f H ] and σ n 2 = [ σ n L 2 σ n H 2 ], respectively. The corresponding numbers of input points are denoted by nL and nH. In accordance with Williams and Rasmussen (2006), Algorithm 1 schematically depicts the implementation of the multi-fidelity GP. Note that the matrix inversion of [ K + σ n 2 I ] is circumvented by using its Cholesky decomposition for the sake of computational efficiency and robustness.
Algorithm 1.

Pseudo-code for the multi-fidelity GP.

Require: f L , f H (input frequencies), y L , y H (observations), cov (covariance function), σ n L 2 , σ n H 2 (noise level), f * (target frequencies) 
L = chol ( K + σ n 2 I ) 
α = L T ( L [ y L y H ] T ) 
evaluate predictive mean h ¯ * = K * α 
β = L K * 
evaluate predictive covariance cov ( h * ) = K * * β T β 
evaluate log likelihood log P ( y | f ) 
  return h ¯ * , cov ( h * ) , log P ( y | f ) 
Require: f L , f H (input frequencies), y L , y H (observations), cov (covariance function), σ n L 2 , σ n H 2 (noise level), f * (target frequencies) 
L = chol ( K + σ n 2 I ) 
α = L T ( L [ y L y H ] T ) 
evaluate predictive mean h ¯ * = K * α 
β = L K * 
evaluate predictive covariance cov ( h * ) = K * * β T β 
evaluate log likelihood log P ( y | f ) 
  return h ¯ * , cov ( h * ) , log P ( y | f ) 

This section involves the boundary element analysis of the vehicle cabin and training procedure of the multi-fidelity GP. Moreover, the results of the multi-fidelity GP are investigated for two objective functions: the SPL and energy density level. In Sec. VI, the error of the multi-fidelity approximation is analyzed.

In this study, the acoustic problem of an industrial vehicle cabin is investigated. The boundary element model of the cabin has been provided by BMW (Munich, Germany Gurbuz , 2022b). The cabin has the following dimensions: 3.032 m length, 1.554 m width, and 1.283 m height. We assume that air is the acoustic medium with the density, ρ a = 1.21 kg / m 3, and the speed of sound, c a = 343 m / s. Two different boundary element meshes of the cabin are studied: A coarse mesh with 1906 degrees of freedom (DOFs) is considered as the LF model, whereas a fine mesh with 24 036 DOFs is adopted as the HF model. In the LF model, boundary elements with constant pressure interpolation are deployed while a linear discontinuous pressure interpolation is used for the HF model (Marburg and Schneider, 2003). The LF cabin model is meshed with an average element edge length of 0.130 m, whereas the average element edge length in the HF mesh amounts to 0.065 m. This leads to 13 elements per wavelength at 200 Hz for the LF mesh and 26 elements per wavelength for the HF mesh. The meshes for the LF and HF models are schematically depicted in Fig. 1.

FIG. 1.

(Color online) Two boundary element meshes for the two fidelity levels. A coarse mesh with 1906 DOFs is adopted as the LF model (a). A finer mesh with 24 036 DOFs is considered as the HF model (b).

FIG. 1.

(Color online) Two boundary element meshes for the two fidelity levels. A coarse mesh with 1906 DOFs is adopted as the LF model (a). A finer mesh with 24 036 DOFs is considered as the HF model (b).

Close modal

To excite the numerical model, we impose a structural particle velocity at the driver's footwell, i.e., v s = 0.001 m / s. The excited boundary elements are visualized in Fig. 2. Regarding the boundary admittance, we impose the Robin boundary condition, ρ a c a Y = f / f ref, with the reference frequency, f ref = 2800 Hz, on each boundary element. In Marburg and Hardtke (1999), the reverberation time was measured in a vehicle cabin at five frequencies between 30 and 300 Hz. By using Eyring's formula, the average absorption coefficient and related boundary admittance were computed. Marburg and Hardtke introduced the reference frequency, f ref = 2800 Hz, because it provided the best approximation for the magnitude of the boundary admittance.

FIG. 2.

(Color online) Excited boundary elements on the LF mesh (a) and HF mesh (b) of the vehicle cabin, where the excitation is modeled as a Robin boundary condition with the real-valued structural velocity, v s = 0.001 m / s.

FIG. 2.

(Color online) Excited boundary elements on the LF mesh (a) and HF mesh (b) of the vehicle cabin, where the excitation is modeled as a Robin boundary condition with the real-valued structural velocity, v s = 0.001 m / s.

Close modal

As customer comfort plays a crucial role in the design of a vehicle cabin, the acoustic quantities are particularly relevant at the position of the driver's ear (Schmid , 2022). A schematic visualization of the field point representing the location of the driver's ear is depicted in Fig. 3. With booming noise being the most prominent problem in the vehicle interior design, the relevant frequency range is chosen to be 20–200 Hz (Luegmair and Schmid, 2020).

FIG. 3.

(Color online) The field point position (red) at the driver's ear from a front view (left) and a lateral perspective (right). For the sake of conciseness, the field point position is only depicted for the HF model.

FIG. 3.

(Color online) The field point position (red) at the driver's ear from a front view (left) and a lateral perspective (right). For the sake of conciseness, the field point position is only depicted for the HF model.

Close modal

Prior to the multi-fidelity analysis, the SPL and sound energy density level are evaluated at the position of the driver's ear. These quantities are computed with the LF and HF models; see Figs. 4(a) and 4(b), respectively. To express the acoustic quantities on a dB scale, SPL = 20 log ( | p | / p ref ) and the sound energy density level, L e = 10 log ( e / e ref ), are computed. For sound waves propagating in air, the reference sound pressure and reference sound energy density adopt to p ref = 2.0 × 10 5 Pa and e ref = 1.0 × 10 12 kg / ms 2, respectively.

FIG. 4.

The SPL (a) and energy density level (b) are evaluated with the LF model (dashed) and HF model (solid). The acoustic quantities are evaluated at the position of the driver's ear.

FIG. 4.

The SPL (a) and energy density level (b) are evaluated with the LF model (dashed) and HF model (solid). The acoustic quantities are evaluated at the position of the driver's ear.

Close modal

The SPL data for the LF model exhibits low values in the frequency range from 20 to 100 Hz with a small peak around 75 Hz. In a previous study (Gurbuz , 2022b), we performed the frequency sweep for the vehicle cabin without damping, e.g., Y = 0. In that study, we observed the first resonance frequency of the cabin at 71 Hz. In Fig. 4, the peak around that frequency is, thus, attributed to the occurence of the first resonance. Above 100 Hz, the SPL at the driver's ear is higher except at 150 Hz, where the SPL decreases to 65 dB [Fig. 4(a)]. Here, again, two additional resonances of the cabin occur at 145 Hz and 170 Hz (Gurbuz , 2022b). As the SPL is evaluated here between two resonances, a drop in the SPL response around 150 Hz becomes natural. Turning now to the sound energy density level with respect to the LF model, we observe a peak at 71 Hz, which is also attributed to the first resonance of the cabin; see Gurbuz (2022b). On average, the energy density level is also higher for frequencies above 100 Hz [Fig. 4(b)]. Around 150 Hz, a small drop in the frequency response becomes apparent. As the energy density level is composed by the potential energy density, which is, in turn, determined by the sound pressure, the same explanation applies to the peak and drop in the energy density level response. Interestingly, we observe very similar SPL and energy density level responses for the HF model. Even though the HF data are generally on a lower level than the data observed in the LF simulation, the profiles of the HF and LF are in good agreement. In addition to a discretization error, the higher average SPL and energy density level for the LF mesh can be attributed to the area of the excited surface. As a matter of fact, the excited surface of the LF mesh, A LF = 0.116 m 2, is significantly larger than the excitation area of the HF mesh, A HF = 0.064 m 2 (Fig. 2). Also, the LF cabin with a length of 2.801 m is slightly shorter than the HF cabin, which is 2.911 m long. By this means, the energy supplied to the LF cabin turns out to be significantly higher than that supplied to the HF cabin. It is worth noting that discrepancies in the transported energy are inevitable because meshes with different fidelity levels provide geometry approximations of varying quality. However, that circumstance is inherently considered within the proposed framework. After all, a positive correlation between the LF and HF data becomes apparent in Fig. 4, which is in accordance with the linearity assumption in Eq. (22).

To implement the multi-fidelity GP, a data selection strategy is required. The strategy proposes the frequencies, f H, at which we evaluate the response function of the HF model, h H. As the simulation of the HF model is accompanied with high computational costs, we are interested in the minimum possible number of HF simulations. At first glance, it might seem adequate to choose data points, which are equidistantly distributed across the frequency range. However, as frequency responses can exhibit regions with resonances or other active events, a more sophisticated approach is required. In this initial study, we introduce an empirical approach. First, the frequencies are identified, at which we observe the maxima in the frequency response and maxima of the curvatures in the LF data. Based on that suggestion, we select a frequency distribution such that we consider the largest possible frequency range. The HF simulation is then performed at the resulting frequencies. This approach is favorable because the LF model is fast to evaluate. Thus, the LF solution is abundant for a wide frequency range, which makes a sophisticated data acquisition strategy irrelevant. As a consequence, we perform the LF BEM simulation at numerous frequencies, which are equidistantly distributed over the entire frequency range.

To assess the performance of the multi-fidelity GP, four cases are investigated in the following:

  • Small LF data set with 37 points, small HF data set with four data points;

  • large LF data set with 181 points, small HF data set with four data points;

  • small LF data set with 37 points, large HF data set with eight data points; and

  • large LF data set with 181 points, large HF data set with eight data points.

By varying the number of relevant data points, we analyze the influence of LF and HF data points on the performance of the proposed method. In the initial study, the noise level, σ n 2 = 0.5 dB, is prescribed on the SPL response of the LF and HF models. Consequently, we obtain a SNR of 20.85 dB in the SPL analysis. Figure 5 shows the results for the multi-fidelity GP with regard to the SPL at the driver's ear. In the initial step, we assign 37 frequencies to the LF model and obtain the HF solution at four frequencies; see Fig. 5(a). Then, the number of LF frequency points is increased to 181 while the HF solution is still known at four frequencies [Fig. 5(b)]. Subsequently, we study the case with the HF solution at 8 frequencies based on the LF solution known at 37 frequencies [Fig. 5(c)]. In the final step, the multi-fidelity GP approximation is investigated for 181 frequency points in the LF solution and 8 points in the HF solution [Fig. 5(d)]. In all of the studies, the multi-fidelity GP approximation is compared to a reference solution. The reference solution is adopted by the HF solution at each frequency. In the initial study with 37 LF and 4 HF frequencies [Fig. 5(a)], one can observe a good agreement between the HF approximation and reference solution. This is particularly the case in the frequency range 20–100 Hz, where the reference solution is either consistent with the HF approximation or at least lies within the 95% confidence interval. Even though we observe slight deviations from the reference solution above 100 Hz, it becomes apparent that the profiles of the HF approximation and reference are similar.

FIG. 5.

(Color online) The SPL at the position of the driver's ear. The LF approximation (blue) of the SPL is evaluated at the LF frequency points (black Y-shaped markers). The HF approximation (orange) of the SPL is evaluated at the HF frequency points (orange crosses). The approximation of the HF model is compared to the reference solution (green). The approximations are associated with 95% confidence intervals (LF, light blue; HF, light orange). The LF and HF data are subject to noise corresponding to SNR = 20.85 dB.

FIG. 5.

(Color online) The SPL at the position of the driver's ear. The LF approximation (blue) of the SPL is evaluated at the LF frequency points (black Y-shaped markers). The HF approximation (orange) of the SPL is evaluated at the HF frequency points (orange crosses). The approximation of the HF model is compared to the reference solution (green). The approximations are associated with 95% confidence intervals (LF, light blue; HF, light orange). The LF and HF data are subject to noise corresponding to SNR = 20.85 dB.

Close modal

When the number of LF frequency points is increased to 181, the HF approximation is even more consistent with the reference solution; see Fig. 5(b). Small deviations are only apparent around 140 Hz. Moreover, we observe that the uncertainty interval is smaller in the LF approximation and the HF approximation is affected in a similar way. By this means, it becomes apparent that the reference solution is not considered within the uncertainties at the frequency range 150–165 Hz. For the remaining frequencies, one can observe that the HF approximation concurs well with the reference. Interestingly, this is also the case between 70 and 170 Hz, even though there is no HF frequency point in it.

In the next phase, the HF system responses are evaluated at 8 frequencies while the LF solution is known at, again, 37 frequencies [Fig. 5(c)]. The additional HF frequency points are placed in the frequency range between 70 and 170 Hz, where we observe an improvement of the HF approximation. This is particularly the case around 130 Hz, where the reference solution is either consistent with the HF approximation or at least included in the uncertainty interval.

In the final step, we analyze the multi-fidelity GP with 181 LF frequency points and eight HF frequency points; see Fig. 5(d). In this study, we observe that the HF approximation is extremely consistent with the reference solution. This is especially the case around 130 Hz, where deviations appeared in the previous studies. Marginal differences, however, occur around 160 Hz, where the HF approximation slightly underestimates the reference solution. This discrepancy can be attributed to a violation of the linearity assumption in the correlation function; see Eq. (22). In particular, this inconsistency may be caused by lags in the frequency responses between the LF and HF response functions. The present results already indicate that the HF solution can be approximated with a drastically reduced number of HF simulations. The approximation can be improved by either adding a few HF frequency points or a great number of LF frequency points. Moreover, the findings show that the uncertainties in the HF approximation can be reduced by involving multiple frequency points. With the LF data being fast to obtain and, thus, abundant, it becomes apparent to prefer a large number of LF solutions for an efficient approximation.

Turning now to the analysis with regard to the sound energy density, the multi-fidelity GP is, again, initialized with 37 LF and 4 HF frequency points. Based on that, the number of frequency points is subsequently increased. In this study, we also prescribe the noise level σ n 2 = 0.5 dB on the LF and HF data, resulting in a SNR of 18.51 dB. The results of the multi-fidelity GP regarding the sound energy density are depicted in Fig. 6. In the initial analysis (37 LF and 4 HF frequency points), the HF frequency points are approximately selected at 20, 80, 140, and 200 Hz, resulting in a nearly equidistant distribution; see Fig. 6(a). For this constellation, the multi-fidelity GP already yields an accurate approximation for the HF model. In particular, the resonance around 70 Hz and the average profile of the sound energy density level are recovered. Small deviations are only apparent between 110 and 150 Hz and 160 and 200 Hz, where the reference solution is overestimated. It is likely that the cause of this discrepancy is a result of the lack of HF solutions within these frequency ranges.

FIG. 6.

(Color online) The total sound energy density level at the position of the driver's ear. The LF approximation of the sound energy density level is evaluated at the LF frequency points (black Y-shaped markers). The HF approximation (orange) of the energy density level is evaluated at the HF frequency points (orange crosses). The approximation of the HF model is compared to the reference solution (green). The approximations are associated with 95% confidence intervals (LF, light blue; HF, light orange). The LF and HF data are subject to noise corresponding to SNR = 18.51 dB.

FIG. 6.

(Color online) The total sound energy density level at the position of the driver's ear. The LF approximation of the sound energy density level is evaluated at the LF frequency points (black Y-shaped markers). The HF approximation (orange) of the energy density level is evaluated at the HF frequency points (orange crosses). The approximation of the HF model is compared to the reference solution (green). The approximations are associated with 95% confidence intervals (LF, light blue; HF, light orange). The LF and HF data are subject to noise corresponding to SNR = 18.51 dB.

Close modal

In the next phase, we increase the number of LF frequency points to 181 while keeping the number of HF points constant [Fig. 6(b)]. At first glance, one can observe an improved approximation for the HF solution. Closer inspection further reveals that the HF approximation agrees well with the reference solution for frequencies between 160 and 200 Hz. Although the profile of the HF approximation and reference resemble each other, we still notice differences between 110 and 150 Hz. Again, these differences are attributed to lags in the frequency responses between the LF and HF models. The fact that the number of LF solutions is increased leads to a decrease in the uncertainties and associated noise levels.

In the third study, we analyze, again, 37 LF points and increase the number of HF points to 8 [Fig. 6(c)]. In comparison to the initial study [Fig. 6(a)], the HF approximation agrees well with the reference solution over the entire frequency range. Small deviations similar to the results in Fig. 6(a) are only apparent above 170 Hz. By including additional HF solutions at 120 and at 170 Hz, the reference solution is considered in the uncertainty interval of the HF approximation even between 110 and 150 Hz and 180 and 200 Hz. The level of uncertainty remains nearly equal compared to the study in Fig. 6(a).

In the final case, the numbers of LF and HF frequency points are increased, i.e., 181 LF and 8 HF points [Fig. 6(d)]. Here, one can observe that the HF approximation is highly consistent with the reference solution, particularly, between 20 and 110 Hz. Marginal deviations only occur around 130 Hz, where the trend of the reference solution cannot be adequately reproduced by the LF solution. Interestingly, we observe that the reference solution does not lie entirely in the uncertainty interval above 140 Hz. There is a marginal discrepancy around 150 Hz, which can be attributed to the presence of noise. The results of the energy density analysis also show that the HF reference solution can be well approximated by a multi-fidelity GP. At closer inspection, one can conclude that including additional HF data improves the approximation. However, the approximation can be similarly improved by solely including additional solutions of the LF model. The involvement of numerous LF solutions should be treated with caution as this approach is associated with the reduction of the uncertainty level. Nonetheless, this approach is highly favorable because a finer resolution of the LF solution provides an efficient way to improve the approximation of the reference solution at lower computational costs. Moreover, our results demonstrate that the proposed method performs very well even in the presence of noise. With the LF model being computationally efficient, the proposed framework provides an efficient method for the approximation of the frequency sweep in boundary element analyses. By this means, accurate and robust predictions can be achieved at lower costs by using multi-fidelity GPs.

To assess the quality of the approximation, the R2 criterion is introduced as (Pesaran and Smith, 1994)
R 2 = 1 ( y * y ̂ H ) T ( y * y ̂ H ) ( y * y ¯ H ) T ( y * y ¯ H ) ,
(37)
where y * is the approximation of the HF frequency response function and y ̂ H is the HF reference solution. The entity, y ¯ H, denotes the mean value of the HF reference solution, which is constantly distributed over the frequency range. For a perfect agreement between the approximation and reference solution, we would obtain R 2 = 1. The resulting R2 values for the predictions of the multi-fidelity GP are depicted in Fig. 7. The results for the SPL analysis show that R2 values around 0.95 are already achieved with two HF points; see Fig. 7(a). By increasing the number of HF points, the R2 criterion even amounts to 0.98. Regarding the LF data, we achieve R2 values close to 0.95 for 60 and 100 LF points. For more than 100 LF points, the R2 criterion rises to 0.98. This result indicates that the influence of the LF points is particularly strong when only a small number of HF points are considered. For more HF points, the performance of the multi-fidelity GP can be significantly improved by including additional LF points. Moreover, this result demonstrates that 4 HF and 100 LF points already suffice to achieve highly accurate predictions.
FIG. 7.

(Color online) R2 values are shown for the predictions of the multi-fidelity GP with regard to the SPL analysis (left) and energy density level analysis (right).

FIG. 7.

(Color online) R2 values are shown for the predictions of the multi-fidelity GP with regard to the SPL analysis (left) and energy density level analysis (right).

Close modal

For the energy density analysis, the R2 value is around 0.90 when we consider only two HF points [Fig. 7(b)]. By adding two further HF points, the R2 criterion rapidly increases to values around 0.98. For more than four HF points, the R2 value remains around 0.98. For 60 LF points, the R2 criterion is around 0.90. By involving 100 or more LF points, one can observe R2 values at 0.98. This finding also shows that 4 HF and 100 LF points suffice for an improved performance. The most interesting result to emerge from the R2 data is that the performance of the multi-fidelity GP is essentially improved by including only a small number of HF points. Based on that, the performance is further improved by adding a decent amount of LF points. As anticipated, discrepancies in the approximation of the HF solution occurred at frequencies where the assumption of a linear correlation [Eq. (22)] was not valid. However, in our studies, they were only marginal; see Figs. 5(d) and 6(d). Next, we observed that the discrepancies could be reduced or at least considered in the uncertainty range by including additional information. This was accomplished by either involving the HF solution at a few more frequencies [Figs. 5(c) and 6(c)] or the LF solution at a decent amount of frequencies [Figs. 5(b) and 6(b)]. Presumably, advanced nonlinear correlation functions would further alleviate this issue.

Table I shows the comparison of the computational runtime between the multi-fidelity GP and HF BEM simulation. For this purpose, the approximation of the frequency sweep is investigated for two cases. The first case refers to the multi-fidelity GP with 37 LF and 4 HF simulations; see Fig. 5(a). In the second case, the frequency sweep is approximated with 181 LF and 8 HF simulations [Fig. 5(d)]. The resulting computational times are compared to the reference method, where the HF simulation is performed at 181 frequencies. As the proposed method requires only a small number of HF simulations, the computational effort is drastically reduced. For instance, in the second case, the runtime is reduced to 3.71 h, resulting in a relative time saving of 92.4%. The computational time is reduced even further in the first case, where the approximation of the frequency sweep is obtained in 1.39 h. In this way, 97.1% of the total runtime is saved. Thus, multi-fidelity GPs provide an efficient technique to overcome the burden of high computational costs and time expense.

TABLE I.

Runtime comparison of the multi-fidelity GPs with the full frequency sweep for the HF BEM model (reference). The multi-fidelity GPs are evaluated for 37 LF points, 4 HF points (case 1) and 181 LF points, 8 HF points (case 2). The runtime for a single LF simulation amounts to 30.45 s, whereas a single HF simulation required 965.04 s.

Multi-fidelity GPs, case 1 Multi-fidelity GPs, case 2 Reference solution
Number of HF simulations  181 
Number of LF simulations  37  181  — 
Multi-fidelity GP training  20.87 s  142.05 s  — 
Total runtime  1.39 h  3.71 h  48.52 h 
Multi-fidelity GPs, case 1 Multi-fidelity GPs, case 2 Reference solution
Number of HF simulations  181 
Number of LF simulations  37  181  — 
Multi-fidelity GP training  20.87 s  142.05 s  — 
Total runtime  1.39 h  3.71 h  48.52 h 

The present study raises the possibility to accelerate frequency sweep studies with the BEM. However, it is important to bear in mind that the present posterior multi-fidelity GP only holds for the position of the driver's ear and cannot be extrapolated to an arbitrary evaluation point. As the transfer functions of the cabin vary for different evaluation points, the multi-fidelity GP needs to be trained again when the position of the evaluation point changes. More specifically, the frequency selection strategy, as introduced in Sec. V B, and the LF model simulations have to be performed from scratch to obtain new proposals for the HF simulations. In contrast to this, the frequency range remains unaffected because it is defined by the underlying problem.

In the sense of a Bayesian method, multi-fidelity models based on GPs account for uncertainties within the data generation process. In the context of the acoustic design of a vehicle cabin, uncertainties are ubiquitous in early design stages. They may occur in the boundary conditions, particularly, in the excitation from the enveloping chassis or boundary admittance of the cabin. Moreover, modifications on the cabin geometry in the development cycle result in variations of the evaluated position. Thus, the position of the driver's ear becomes an additional source of uncertainty. Uncertainties can further occur as a result of simplifying model assumptions. For the Helmholtz equation, this may involve the assumption that the underlying acoustic problem can be considered to be linear.

Finally, it should be noted that our findings are conducted with a squared exponential covariance function. Among the vast amount of covariance functions, complex-valued and physics-based covariance functions (Caviedes-Nozal , 2021) sound very promising as they account for spatial information on the characteristics of the propagating waves.

This paper set out to improve the efficiency in the design of acoustic systems. Therefore, a multi-fidelity model was developed based on boundary element simulations of two different meshes. A coarse resolution of the mesh was adopted as the LF model, whereas a fine mesh was considered as the HF model. The fidelity levels were realized by GPs, which were trained with the related frequency responses. To demonstrate the effectiveness of our method, we investigated a vehicle interior noise problem, where we analyzed the frequency response of the cabin with respect to the driver's ear. Regarding the objective function, the SPL and sound energy density at the driver's ear were used. In our study, we compared the approximation of the multi-fidelity GP with the HF solution at each frequency. Moreover, we assessed the performance of the proposed method by studying the influence of LF and HF solutions at additional frequencies. The results in our study have shown that the HF solutions were efficiently approximated by the proposed method. In addition to this, sensitivities to model parameters and uncertainties arising from the data generation process were inherently quantified. By this means, multi-fidelity GPs provide an efficient and robust tool to determine the frequency-dependent characteristics of acoustic systems. The evidence from our work, thus, emphasizes the exploration of multi-fidelity GPs in acoustic problems. With a rather semi-empirical approach to determine the relevant frequencies, a natural progression of this work is to implement a more elegant technique for the selection of frequency points at which the HF model is evaluated. One possible strategy could involve a decision criterion in the form of a loss function. Future work needs to examine the links between the HF model and LF model because we only focused on linearly correlated models in this study. Moreover, we strongly recommend the integration of different data generation sources in the proposed framework. This involves particularly physical experiments as well as analytical solutions within a model of multiple fidelity levels.

This research was funded by the German Research Foundation (DFG) under Award No. 418936727. Moreover, we are grateful to Dr. Marinus Luegmair for providing the model of the vehicle cabin and fruitful discussions.

Following Williams and Rasmussen (2006), the joint probability distribution of two Gaussian random vectors, x and y, reads
[ x y ] N ( [ μ x μ y ] , [ A C C T B ] ) .
(A1)
The conditional probability distribution for x given y, then, can be expressed by
P ( x | y ) = N ( μ x + C B 1 ( y μ y ) , A C B 1 C T ) .
(A2)
In our study, the observations are stored in y while the unknown function values are expressed by x.
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