Surrogate-Based Uncertainty Quantification for Coupled Structural–Acoustic Problems

Abstract

This paper presents a surrogate-based uncertainty quantification (UQ) framework for coupled structural–acoustic systems subject to material and geometric variability. The proposed methodology integrates the Finite Element Method (FEM) with two metamodeling techniques—the Quadratic Response Surface (QRS) and Kriging—and Monte Carlo Simulations (MCS), to efficiently characterize the probabilistic behavior of the acoustic response. Two accuracy metrics (cross-validation error and prediction error) are used to validate the surrogate models. Numerical experiments demonstrate that the Kriging metamodel trained with 30 Latin Hypercube Sampling (LHS) points achieves superior predictive accuracy, with a Relative Maximum Error of 4.125 × 10⁻⁷. Monte Carlo Simulations conducted via the Kriging surrogate reduce the computational cost by more than six orders of magnitude compared to direct FEM-based MCS, while maintaining high accuracy. The proposed framework is validated on a rectangular cavity coupled with two flexible aluminum plates, and provides an efficient and accurate tool for vibro-acoustic UQ in complex engineering systems.

1. Introduction

Researchers in a variety of domains, including car compartments, airplane cabins, etc., have been devoted to solving the complex coupled physics problem of vibrating elastic structures interacting with confined acoustic cavities. An excellent review was given by the authors of [1,2], which examined the interacting vibration–acoustic mechanisms within these systems, including the modal coupling mechanisms between interconnected structural and acoustic domains. In [3], the authors suggested a sophisticated technique for numerically simulating intricate noise problems, dubbed the acoustic hybrid approach. An approximate resolution to the coupled systems’ response can be achieved by exploiting the Finite Element Method (FEM), which is a very effective technique [4,5].

The Finite Element Method (FEM) has been widely employed for the numerical analysis of coupled structural–acoustic systems, owing to its versatility and well-established accuracy [4,5,6,7,8]. Alternative simulation strategies have also been proposed in the literature, including the Boundary Element Method (BEM) [9,10], Isogeometric Analysis (IGA) [11,12], and high-order hp-FEM formulations [13]. In the present study, standard FEM is retained as it provides an effective balance between accuracy and computational tractability for the bounded cavity problem considered.

Numerical analysis of the vibration–acoustic field’s frequency response has garnered a lot of attention, without taking into account the uncertainty related to various parameters, as is generally known. However, in order to guarantee the robustness and dependability of the findings, this study incorporated several sources of uncertainty, including manufacturing defects, material properties, boundary conditions, etc. [14,15]. From the literature, we were able to categorize uncertainty analysis approaches into two primary groups: probabilistic approaches [16,17,18] and non-probabilistic approaches [19,20]. Of all the probabilistic methods, statistical methods or sampling strategies, such as Monte Carlo Simulations (MCSs) [21,22], are the most commonly utilized. For these methods to achieve a respectable level of accuracy, several samples of the entered variables are needed. For every realization, the issue is then resolved. Nonetheless, there should be a sufficient number of realizations, i.e., ${10}^5$ or ${10}^6$ deterministic FEM must be used in order to obtain precise findings.

In general, it becomes impractical to run numerous simulations for quantifying uncertainty [23,24] or design optimization [25] when employing computationally costly simulation codes in complicated engineering issues. A superior substitute is found in the application of original model approximations, commonly referred to as meta-models (or surrogate models). These models seek to define the relationship between certain system inputs and outputs by using mathematical models [26,27,28]. Kriging [22,29] and Response Surface Methodology (RSM) [30,31] are the two most widely used meta-modeling techniques.

This study conducts a frequency-domain analysis of coupled structural–acoustic systems, incorporating uncertainties in both geometric dimensions and material properties. Two surrogate modeling approaches—Response Surface Methodology and Gaussian Process Regression (Kriging)—are comparatively evaluated for their ability to capture input–output relationships in high-fidelity simulations. The research further implements Monte Carlo Simulation through the Kriging surrogate to statistically assess vibro-acoustic performance under parametric variability.

2. Uncertainty Quantification Analysis Using Surrogate Models

The MCS technique incorporates uncertain parameters specified by random variables into consecutive resolutions of a deterministic system. It creates random samples for any unclear parameter. The fundamental benefit of this method is that it may be used on systems of varying scales and complexity (linear, non-linear, etc.). Many points are needed to provide a good level of precision in the results.

Surrogate Modeling Techniques

Surrogate modeling techniques are increasingly employed in contemporary engineering practice across various disciplines. These computational methods create efficient approximations of complex finite element analyses, capturing the functional relationships between system inputs and performance outputs.

For an m-dimensional design issue, we define $\mathit{x}=\left(x_1,x_2,\dots ,x_m\right)$ as the design variable vector, ${\mathit{x}}_S={\left({\mathit{x}}^{\left(1\right)},{\mathit{x}}^{\left(2\right)},\dots ,{\mathit{x}}^{\left(n\right)}\right)}^T$ as the experimental design matrix containing n sample points, and ${\mathit{y}}_S={\left(y\left({\mathit{x}}^{\left(1\right)}\right),y\left({\mathit{x}}^{\left(2\right)}\right),\dots ,y\left({\mathit{x}}^{\left(n\right)}\right)\right)}^T$ as the corresponding system responses. The dataset (X, y) forms the basis for constructing a predictive model capable of estimating responses ŷ(x) at unsampled locations. Surrogate model development follows a systematic workflow: design space exploration via sampling techniques; high-fidelity model evaluations at sample points; surrogate model construction; and predictive accuracy validation. Latin Hypercube Sampling (LHS), introduced by [27], provides an efficient space-filling sampling strategy, after which appropriate surrogate models are fitted to the acquired data [22,28].

3. Design Optimization Under Reliability

The eventual purpose of the design under uncertainty is to maximize the overall cost. In theory, it is necessary to achieve an ideal compromise between the reliability of the structural system and other, sometimes conflicting, societal goals. This is a tough mission. Usually, the resolution of the Reliability-Based Design Optimization (RBDO) model is attained by combining optimization repetitions and reliability [18,32]. This method produces poor numerical effectiveness, which is inconvenient for engineering utilization on actual structures. For the purpose of evading this complication, hybrid RBDO methods are suggested [33,34]. Similarly, in this study, we suggest the dynamic hybrid method [35,36]. The effectiveness of this approach is indicated in [16,22].

3.1. Dynamic Hybrid Method

The resolution of the over-nesting issue requires an extremely long computation time, primarily for extensive structures. To enhance numerical efficiency, the hybrid method minimizes a reformulated objective function $F\left(\left\{x\right\},\left\{y\right\}\right)$, combining deterministic x and random y variables as follows:

$$f\left(\left\{x\right\}\right)·d_{\beta }(\left\{x\right\},\left\{y\right\})$$

(1)

This method is constrained by the following limit state constraint:

$$\begin{gathered} G\left(\left\{x\right\},\left\{y\right\},t\right)\le 0 \forall t\in \left[0 T\right] \\ {d}_{\beta }\left(\left\{x\right\},\left\{y\right\},t\right)\le {\beta }_t \left(t\right) \end{gathered}$$

(2)

and the deterministic and reliability constraints:

$$g_k\left(x,t\right)\le 0$$

(3)

For the static problem case (t = 0), dβ({x},{y}) denotes the shortest distance, measured in the normalized hybrid design space (HDS), from the current point ({x},{y}) to the limit-state surface G({x},{y}) = 0. The reliability constraint dβ({x},{y}) ≤ β_t ensures that the design operates within the feasible region corresponding to the target reliability index β_t. When the random variables follow a normal distribution, this distance equals the Hasofer–Lind reliability index, and the iso-reliability contours take the form of ellipses in the HDS, as illustrated in Figure 1.

Figure 1. HDS illustration in the case t = 0. Solid curves: objective function levels f(x); dashed lines: limit state functions G(x,y); elliptical contours: iso-reliability curves (critical contour at d_β = β_t); P_x: deterministic optimal point (d_β < β_t); P_y: reliability design point on G(x,y) = 0 and d_β = β_t; superscript * denotes optimality.

Figure 1 illustrates this HDS through an example, integrating both types of variables. When variables follow a normal distribution, the reliability thresholds $d_{\beta }$ can be represented as elliptical contours. The solid curves correspond to the objective function’s levels, whereas the dashed lines depict the limit state functions (except for $G\left(\left\{x\right\},\left\{y\right\}\right)=0$). Two critical resolutions emerge from this representation, namely the optimal resolution $P_x^{*}$ and the reliability resolution $P_y^{*}$, the latter being the design point located on the $G\left(\left\{x\right\},\left\{y\right\}\right)=0$ and $d_{\beta }={\beta }_t$ curves.

3.2. Particle Swarm Optimization

Particle Swarm Optimization (PSO) was first introduced in 1995 by James Kennedy and Russell C. Eberhart. This method is inspired by the social behavior of swarms, where individual agents, referred to as particles, interact within a collective group (or swarm) to achieve a common goal in a shared search space. Each particle possesses memory and information-processing capabilities [37,38]. PSO is the optimization engine envisioned for the RBDO phase of this research study. Each particle updates its position based on three components: inertia (tendency to continue in the current direction), cognitive influence (attraction to the particle’s personal best), and social influence (attraction to the global best found by the swarm) [39]. A flowchart of the PSO algorithm is illustrated in Figure 2.

Figure 2. Flowchart of the classical PSO algorithm.

Over time, PSO techniques have undergone significant advancements, with modern variants bearing little resemblance to the original algorithm. In this work, we opt for a Parallel PSO (PPSO) model, an enhanced version of the standard PSO algorithm. The PPSO integrates an adaptive neighborhood mechanism combined with parallel computing to improve performance.

3.3. Parallel Particle Swarm Optimization

PSO has emerged as one of the most effective metaheuristic algorithms for solving complex optimization problems. Despite its widespread adoption and robustness, PSO suffers from notable limitations, including high computational costs and a tendency toward premature convergence.

To address these challenges, the Parallel PSO (PPSO) model was developed as an advanced variant of the standard PSO algorithm. PPSO introduces an adaptive neighborhood mechanism, which enhances search diversity and improves exploration of the solution space. This innovation reduces the risk of premature convergence while maintaining the ability to reach globally optimal solutions [39].

However, neighborhood-based approaches can still be computationally expensive, particularly for large-scale or complex optimization problems. To mitigate this, PPSO leverages parallel computing to accelerate processing and deliver optimal solutions within a reduced timeframe. By parallelizing the entire PSO workflow, computational efficiency is significantly improved. Figure 3 illustrates the flowchart of the proposed PPSO model.

Figure 3. Schematic diagram of the parallel PSO algorithm.

The integration of PPSO with the Kriging surrogate for full RBDO application will be presented in a follow-up publication; the present manuscript focuses exclusively on the surrogate-based UQ framework.

4. Numerical Results

The method employs the deterministic FEM to compute responses at training points (generated via Latin Hypercube Sampling (LHS)), followed by MCS-based uncertainty analysis using a Kriging surrogate model.

4.1. Deterministic Analysis Using Finite Elements

The system under study consists of two flexible aluminum plates (Plate 1: thickness 2 mm; Plate 2: thickness 4 mm), shown in Figure 4 [15]. The acoustic environment features a rigid-walled cavity, creating a coupled structural–acoustic interaction problem. Numerical modeling was performed using ANSYS-APDL 17.2 with the fluid domain discretized into 3000 elements (eight-node Fluid30 elements with acoustic pressure as a single degree of freedom) and plate structures meshed with 200 elements (four-node Shell181 elements featuring six DOFs per node: translations and rotations about x, y, z axes) (see Figure 5). The mesh density was selected to ensure a minimum of six elements per acoustic wavelength at the maximum analysis frequency of 1000 Hz (corresponding to a wavelength of approximately 0.34 m in air). This criterion, standard in FEM acoustics, is satisfied by the 3000-element Fluid30 mesh. For the plate structures, the 200-element Shell181 mesh captures at least the first ten structural modes within the frequency range of interest. Boundary conditions include full clamping constraints along all plate edges. The aluminum plates are modeled with Young’s modulus E = 70 GPa, density ρ = 2700 kg/m³, and Poisson’s ratio ν = 0.33.

Figure 4. An acoustic–structural model.

Figure 5. Ansys FEM model. The red arrow indicates the applied unit harmonic force at excitation point F, located at the center of Flexible Plate 2.

Harmonic analysis covers the frequency range from 0 to 1000 Hz with a step of 1 Hz. A unit harmonic force (1 N) is applied at point F at the center of Flexible Plate 2. Structural damping is modeled using a constant hysteretic loss factor η = 0.01 applied to both aluminum plates; fluid damping is neglected, consistent with the rigid-wall cavity assumption. Fluid–structure coupling is enforced at all plate free-face nodes via the ANSYS APDL coupled-field formulation (FLUID30–SHELL181 interface). The monitoring point G2 is located at specific coordinates (0.05 m, 0.05 m, 0) along the bottom wall of the acoustic cavity, corresponding to the geometric center of the cavity base plane.

The FEM is utilized to analyze and compute the acoustic pressure distribution, with particular focus on the Sound Pressure Level (SPL) at monitoring location G2, as illustrated in Figure 6.

Figure 6. Internal acoustic cavity sound pressure level. The red line represents the Sound Pressure Level (SPL) frequency response computed at monitoring point G2 using the deterministic FEM analysis.

4.2. Proposed Surrogate-Based Uncertainty Analysis

This study treats the aluminum plates’ material properties (elastic modulus and density), along with their thickness dimensions, as probabilistic variables. The statistical characteristics of these four uncertain parameters are detailed in Table 1.

Table 1. Characteristics of random variables.

Physical Property	Description	Nominal Value	Variability (COV)	Statistical Distribution
E (GPa)	Young’s modulus for both plates	70	5%	Normal
ρ (kg/m3)	Density for both plates	2700	5%	Normal
t1 (mm)	Thickness of Plate 1 (side plate)	2	2%	Uniform
t2 (mm)	Thickness of Plate 2 (top plate)	4	2%	Uniform

The probability distributions assigned to the uncertain parameters reflect standard engineering assumptions. Material properties (E, ρ) are modeled as normal random variables with a coefficient of variation (COV) of 5%, which is consistent with the reported manufacturing variability for structural aluminum alloys. Plate thicknesses (t1, t2) are modeled as uniform random variables with a tighter COV of 2%, reflecting the bounded dimensional tolerances of sheet metal forming (ISO 2768-1 standard [40]), where values within the tolerance band are equally probable. All four random variables are assumed to be statistically independent, as no empirical correlation data between material and geometric parameters were available for this study.

The peak sound pressure level within the acoustic cavity is computed through two distinct surrogate modeling techniques: QRS and Kriging interpolation. Kriging implementation employs MATLAB R2019a’s DACE toolbox [41], configured with a quadratic global trend component that has been paired with an exponential covariance kernel for spatial correlation. Separate Latin Hypercube Sampling points generate independent training data for each surrogate model to assess their predictive capabilities, while additional validation points evaluate model fidelity. Finite Element Method simulations are iteratively conducted using the training dataset specified in Table 1. Model accuracy is rigorously quantified through three standard metrics—Mean Absolute Error (MAE), Relative Maximum Error (RME), and Root Mean Square Error (RMSE)—as presented in Table 2. The cross-validation results comparing 20-point and 30-point LHS designs are visualized in Figure 7 and Figure 8, respectively, demonstrating the influence of sampling density on model performance. The LOO cross-validation procedure withholds each training point in turn, retrains the Kriging model on the remaining n-1 points, and records both the predicted value and the Kriging predictive standard deviation at the withheld point. For the 30-point design, the mean LOO predictive standard deviation is 8.93 × 10⁻³ dB, confirming that the surrogate’s local uncertainty is negligible relative to the SPL range.

Table 2. Accuracy comparison of surrogate methods.

Error Measure	20-Point LHS		30-Point LHS
	QRS	Kriging	QRS	Kriging
MAE	0.656	2.255 × 10−4	0.589	1.269 × 10−4
RME	0.003	5.191 × 10−7	0.002	4.125 × 10−7
RMSE	0.013	5.851 × 10−5	0.002	8.926 × 10−6

Figure 7. Cross-validation outcomes of SPL using 20-point LHS.

Figure 8. Cross-validation outcomes of SPL using 30-point LHS.

The comparative analysis in Table 2 demonstrates the superior performance of the Kriging surrogate model compared to the QRS approach. Cross-validation analysis reveals enhanced predictive accuracy when using the 30-point LHS with the Kriging method. Based on these favorable outcomes, the Kriging model has been selected for subsequent Monte Carlo uncertainty quantification. For this analysis, a sample size of $N={10}^6$ was generated for the random variables (Table 1 parameters), with MCS executed through the Kriging surrogate. The resulting mean sound pressure level distribution is illustrated in Figure 9.

Figure 9. Mean value of the SPL obtained using the Kriging-based MCS approach (N = 10⁶ samples).

The dominant source of output variability in a coupled structural–acoustic system near resonance is the sensitivity of the natural frequency fₙ to material and geometric uncertainties. First-order uncertainty propagation was used as follows:

$$\sigma (fₙ) \approx fₙ \times \sqrt{{(0.5\times COV\_E)}^2 + {(0.5\times COV\_\rho )}^2 + {(0.5\times COV\_t_1)}^2 + {(0.5\times COV\_t_2)}^2}$$

This yields σ(fₙ) ≈ 25 Hz for the present parameter set. Since the resonance half-bandwidth is Δf ≈ 6.8 Hz, the standard deviation of the resonance frequency substantially exceeds the half-bandwidth. Each MCS realization therefore places the resonance peak at a different frequency, producing a very wide SPL distribution near resonance and near-zero variance off resonance. The MCS output is characterized by a high-variance region in the resonance band (~640–720 Hz), where the 5th–95th percentile band spans approximately 70–115 dB, and negligible variance at off-resonance frequencies. With N = 10⁶ samples, the mean estimator error scales as ${\displaystyle \frac{\sigma }{\sqrt{N}}}\approx 0.013 \mathrm{dB}$, confirming full statistical convergence.

The computational requirements merit special consideration. A single deterministic FEM simulation requires approximately 3 min of processing time. For the training phase involving 30 sample points, this translates to 90 min of total computation (30 × 3 min). Notably, a full Monte Carlo Simulation using direct FEM would demand approximately 3 million minutes ($3\times {10}^6$) for complete uncertainty analysis.

In contrast, the Kriging-based MCS approach demonstrates remarkable efficiency, completing the analysis in just 27 s. This substantial reduction in computational expense—from millions of minutes to mere seconds—clearly establishes the superior efficiency of the surrogate model approach compared to conventional FEM-based Monte Carlo methods.

5. Conclusions

This study demonstrates the effectiveness of surrogate-based uncertainty quantification for coupled structural–acoustic systems. The principal findings are as follows: (1) Accuracy: Among the two surrogate families evaluated, Kriging consistently outperforms QRS. With 30 LHS training points, Kriging achieves a Relative Maximum Error of 4.125 × 10⁻⁷ and an RMSE of 8.926 × 10⁻⁶, representing a reduction of approximately three orders of magnitude compared to QRS. (2) Training set size: Cross-validation results for this specific four-dimensional benchmark problem show that 30-point LHS provides better predictive accuracy than 20-point LHS. This suggests that 30-point LHS represents a reasonable starting point for problems of similar dimensionality and smoothness. (3) Computational efficiency: The surrogate-based MCS (N = 10⁶ samples) is completed in 27 s, compared to an estimated 3 × 10⁶ min for direct FEM-MCS—an increase exceeding 6 × 10⁶. The total computational budget, including FEM training simulations, amounts to 90 min. (4) Scope and limitations: The present study is validated on a single benchmark case. Generalization of the proposed framework to industrial problems with higher dimensionality, multiple response quantities, or non-stationary responses should be approached with appropriate validation. An extension to such cases is proposed as future work.

For vibro-acoustic problems with sharp resonance peaks, Kriging is recommended over polynomial surrogates due to its local interpolation property and built-in predictive uncertainty. The proposed framework provides a validated methodological basis for vibro-acoustic UQ, with extension to more complex industrial configurations as a recommended next step.