Successive subdivisions of the initial mesh produce a smooth limit surface. Within subdivided surfaces, a triangular element is defined as regular if all its vertices are regular (valence nv=6) [34] (see the red triangular element in Fig. 4). For a point xe with local coordinates (θ1,θ2) (θ1,θ2∈[0,1]) in the regular element, its global coordinates can be directly interpolated using the quartic box-spline basis functions Bi and the coordinates of the corresponding control points xi (i.e., the 12 surrounding vertices):

(3)xe(θ1,θ2)=∑i=112Bi(θ1,θ2)xi.

For regular elements, Stam [35] defines the explicit form of these basis functions, which form the foundation for subdivision surface parameterization (Eq. (3)). (4)B1=112(θ14+2θ13θ2),B2=112(θ14+2θ13θ3),B3=112(θ14+2θ13θ3+6θ13θ2+6θ12θ2θ3+12θ12θ22+6θ1θ22θ3+6θ1θ23+2θ23θ3+θ24),B4=112(6θ14+24θ13θ3+24θ12θ32+8θ1θ33+θ34+24θ13θ2+60θ12θ2θ3+36θ1θ2θ32+6θ2θ33+ 24θ12θ22+36θ1θ22θ3+12θ22θ32+8θ1θ23+60θ23θ3+θ24),B5=112(θ14+6θ13θ3+12θ12θ32+6θ1θ33+θ34+2θ13θ2+6θ12θ2θ3+6θ1θ2θ32+2θ2θ33),B6=112(2θ1θ23+θ24),B7=112(θ14+6θ1θ33+12θ12θ32+6θ1θ33+θ34+8θ1θ2+36θ12θ2θ3+36θ1θ2θ32+8θ2θ33+24θ12θ22+ 60θ1θ22θ3+24θ22θ32+24θ1θ23+24θ23θ3+6θ24).

For irregular elements consist of irregular vertices (nv≠6), the direct interpolation for points in regular elements is no longer applicable. To this end, a feature-based subdivision matrix method is adopted, whereby the irregular element is iteratively refined until the fitting point is relocated into a regular sub-element. Fig. 5 illustrates this process for a red triangular element containing vertex 1 (nv≠6), which is subdivided into three regular sub-elements (Γ11,Γ21 and Γ31) and one irregular sub-element. If the fitting point xe falls into one of the three regular sub-elements, then its parametric coordinates can be computed using the direct interpolation (see Eq. (3)). If xe falls into the irregular sub-element, this sub-element will be subdivided iteratively until xe locates in a regular sub-element. Detailed numerical implementation of the subdivision matrix method can refer to reference [35], Hence, the point in the irregular element can also be interpolated as:

(5)xe(θ1,θ2)=∑i=1nv+6Ni(θ~1,θ~2)xi,which uses nv+6 control points, and the local coordinates in sub-elements (θ~1,θ~2)=(θ1,θ2)/2l, and l denotes the level of subdivision. More details can refer to [36].

As shown in Fig. 7, the Conditional Generative Adversarial Network (CGAN) [45] comprises two adversarial neural networks: a generator that produces synthetic data and a discriminator that differentiates between real and generated data. In this study, the generator G receives random noise Z (a vector of random values between 0 and 1) and conditional input Y (representing frequency, e.g., 100 Hz) as input data, and produces (i.e., outputs) synthetic data G(Z,Y) that mimics the statistical distribution of real data. By integrating the conditional information Y, the model shifts from an unsupervised to a supervised learning paradigm, which facilitates precise control over the generated outputs.

The discriminator D distinguishes real data from synthetic data, and thus its inputs include the real sound pressure data X (sound pressure values) with its corresponding condition Y (frequency), and the generated data G(Z,Y) paired with the same condition Y, while its output will be a binary value (0 or 1) denoting whether the data is real or synthetic data. During training, the generator and discriminator engage in an adversarial process, iteratively improving their performance. This dynamic resembles a minimax game, where both networks progressively converge toward a Nash equilibrium. At this equilibrium, the generator produces data indistinguishable from real data, and the discriminator cannot reliably differentiate between the two. Finally, the Table 1 presents the input and output structure of the CGAN model. In the CGAN model, the generator takes as input the random noise Z and the conditional information Y (i.e., frequency), and outputs G(Z,Y), which represents the generated data that follows the same pattern and distribution as the sound pressure. The discriminator receives the real sound pressure data X along with the corresponding condition Y, and outputs a binary result (0 or 1) indicating whether the input data is real or generated. Table 1 presents the input and output structure of the CGAN model.

Eq. (19) defines the CGAN’s adversarial objective function, which incorporates conditional information to establish a minimax optimization framework: (19)minGmaxDV(D,G)=EX∼Preal(X)[log⁡D(X,Y)]+EZ∼PZ(Z)[log⁡(1−D(G(Z,Y),Y))].

Here, V denotes the value function of the adversarial game, and E represents the expectation operator. D(X,Y) corresponds to the discriminator’s output probability when evaluating real data X paired with its associated condition Y. Preal(⋅) and PZ(⋅) denote the probability distributions of the real data and latent noise, respectively. D(G(Z,Y),Y) reflects the discriminator’s evaluation of the generator’s synthetic output G(Z,Y) taking into consideration the conditional input Y.

In this architecture, the discriminator D operates as a binary classifier tasked with distinguishing real data from synthetic samples, whereas the generator G aims to synthesize data that matches the statistical distribution of the real data Preal(⋅). By explicitly conditioning both networks on Y, CGAN addresses the inherent limitations of traditional Generative Adversarial Networks (GANs), such as uncontrolled generation, and enables targeted synthesis of outputs aligned with specific frequency-based constraints.

During adversarial optimization, the generator G and the discriminator D iteratively refine their performance until reaching an equilibrium state where the discriminator cannot reliably differentiate between real and synthetic data. The training alternates between updating G and D, with the global loss function L(D,G) unifying their individual objectives: (20)L(D,G)=EX,Y∼Preal(X,Y)[log⁡D(X,Y)]+EY∼PY(Y),Z∼pZ(Z)[log⁡(1−D(G(Z,Y),Y))],while the loss functions for D and G can be expressed as: (21)LD=−E(X,Y)∼Preal(X,Y)[log⁡D(X,Y)]−EZ∼PZ(Z),Y∼PY(Y)[log⁡(1−D(G(Z,Y),Y))]LG=EZ∼PZ(Z),Y∼PY(Y)[log⁡(1−D(G(Z,Y),Y))].

Notably, CGAN’s generator inherently produces diverse synthetic datasets by sampling from Preal(⋅) and PZ(⋅). These generated samples further refine the discriminator’s accuracy during training, creating a mutually reinforcing cycle. This capability enables CGAN to address the challenges outlined in this study, such as generating targeted outputs under conditional constraints while maintaining data diversity.

We employ the boundary element method based on subdivision surfaces to analyze a spherical object with a radius of 1.2 m subjected to a unit-amplitude plane wave incident along the x-axis, the wave number is 0.100. The observation point is located at the coordinates (3, 0, 0 m), and additional geometric details are illustrated in Fig. 9. As a canonical three-dimensional acoustic problem [46], this spherical model has an analytical solution, making it a reliable benchmark for evaluating the accuracy and efficiency of the proposed algorithm.

The spherical mesh was refined iteratively using the Loop subdivision surface method (Fig. 10). Starting from a coarse polyhedral mesh, successive subdivision steps generated progressively smoother surfaces. Thus, it can bring higher computational accuracy.

The Table 2 presents the errors between the numerical solutions obtained by the Boundary Element Method (BEM) and the analytical solutions for the spherical model with an initial mesh of 768 elements after two successive refinements. It can be observed that the refined mesh obtained through subdivision exhibits significantly smaller errors between the numerical and analytical solutions in the low-frequency range compared to the coarse mesh, resulting in an overall improvement in computational accuracy relative to the unrefined mesh. The spherical model in the example is based on an initial mesh of 768 elements, refined two times.

The discretized spherical model was solved via the Boundary Element Method (BEM), and a CGAN model was then trained to predict acoustic responses. To validate the proposed method’s correctness, an initial dataset generated by BEM calculations was divided into training and testing sets, with the CGAN training process detailed in Appendix A. After training, the model’s accuracy was evaluated on the test set, as shown in Figs. 11 and 12.

Fig. 11 compares the real and imaginary components of the acoustic pressure predicted by CGAN, DNN, and conventional models, while Fig. 12 illustrates their performance in predicting sound pressure magnitude. The CGAN model outperforms other methods in capturing both the complex components and overall pressure trends. By incorporating generator-synthesized data during training, CGAN maintains high accuracy even at dataset boundaries (100 and 400 Hz), where DNNs [47–49] struggle due to sparse boundary data. Notably, CGAN predictions align closely with analytical solutions, achieving accuracy comparable to the conventional BEM (CBEM).

The subdivision surface method enhances computational accuracy by iteratively refining meshes. However, this refinement significantly increases mesh density, leading to higher computational costs and prolonged simulation times. Consequently, traditional boundary element methods (BEM) struggle to efficiently handle complex models with fine meshes due to these scalability limitations. To mitigate this trade-off, the CGAN model was introduced to augment sparse datasets and accelerate computations.

To do this, the conventional boundary element method (CBEM) first computed sound pressure values at the observation point (3, 0, 0) across the 100–400 Hz frequency range using coarse 10 Hz intervals. This sparse dataset served as the training input for the CGAN model. After training, the CGAN generated predictions at a refined 1 Hz resolution, effectively augmenting the dataset. The augmented results were then rigorously validated against both CBEM outputs and analytical solutions, as demonstrated in Figs. 13 and 14, confirming the method’s ability to balance accuracy and efficiency.

The CGAN-augmented data shows a high degree of agreement with both the analytical solution and the CBEM results, confirming that the model successfully learned the underlying data patterns and generalized effectively to unseen frequencies. As shown in the subfigure, relative errors at the dataset boundaries (100 nd 400 Hz) with respect to both algorithms are less than 2%. Specifically, the Fig. 14 demonstrates that CGAN and CBEM achieve nearly identical accuracy, as the errors are very small. These findings validate CGAN as an effective tool for accelerating computations without sacrificing precision. Furthermore, comparison with CBEM results reveals that CGAN achieves comparable predictive accuracy while significantly reducing computational effort, establishing its dual advantage in both accuracy and efficiency for acoustic analysis.

Table 3 provides a validation of the CGAN model’s predictive accuracy in the mid-frequency range, confirming that CGAN and CBEM achieve statistically indistinguishable error levels relative to analytical solutions, with discrepancies below 0.5% in mid-range frequencies. However, CGAN reduces computational time by orders of magnitude compared to CBEM when processing equivalent datasets. Typically, training a well-performing CGAN model takes about 30 min, while generating the initial dataset through numerical simulation with a frequency step of 10 Hz requires approximately 35 min. Overall, compared to computing all frequency points, this approach reduces computational costs. However, in practical applications, it is usually unnecessary to calculate the sound pressure at every frequency point. This condition is set here mainly to verify that CGAN can effectively assist computations in certain specific cases.

From this example, it can be observed that the subdivision surface method further enhances computational accuracy by generating smooth, high-quality meshes through iterative refinement of coarse polyhedral models. This approach produces multi-resolution meshes capable of handling complex geometries while improving numerical integration accuracy—a critical factor for precise boundary element calculations. Its adaptability to intricate structures, coupled with widespread software integration, makes it indispensable for 3D acoustic problems requiring geometric fidelity. However, finer meshes increase computational costs, exacerbating the trade-off between accuracy and efficiency.

This challenge is mitigated by the CGAN model, which uniquely addresses data scarcity inherent to high-fidelity 3D acoustic modeling. Conventional neural networks struggle in such scenarios due to the prohibitive computational cost of generating large training datasets. CGAN circumvents this limitation by leveraging conditional information (e.g., frequency) to synthesize physically realistic data through its adversarial framework. The generator produces virtual data distributions that mimic real data, while the discriminator refines its validation criteria iteratively. This process enables robust training on sparse datasets, achieving accuracy rivaling CBEM with far less computational overhead. By integrating subdivision surfaces for computational accuracy and CGAN for data-efficient acceleration, the proposed framework offers a balanced solution to the accuracy-efficiency trade-off in 3D acoustic analysis.

To validate the generalizability of the CGAN framework across complex geometries and diverse acoustic scenarios, we analyzed a washing machine model in this section. The observation point was positioned at (0.5, 0, 0 m), with geometric and boundary condition details illustrated in Fig. 15. First, we considered a unit-amplitude plane wave incident along the positive x-axis, analogous to the spherical model case.

The model was discretized using Loop subdivision surfaces, with mesh refinement levels shown in Fig. 16 and quantified in Table 4. Each subdivision level quadruples the number of nodes and elements, progressively approximating the ideal subdivided surface. However, higher subdivision levels impose prohibitive computational and memory demands, rendering BEM calculations impractical. To balance accuracy and resource constraints, we selected the Level 1 mesh (43,658 nodes, 87,272 elements) for subsequent analyses.

BEM simulations for the washing machine model proved computationally prohibitive, with even sparse initial datasets requiring substantial calculation time. To address this, we implemented a two-stage acceleration strategy. First, the fast multipole method (FMM) was integrated with the Loop subdivision surface technique to create an accelerated BEM framework (FMBEM) for efficient initial data generation.

The accuracy of this FMBEM implementation was validated by comparing real and imaginary sound pressure components at the observation point against conventional BEM results, as illustrated in Fig. 17. The near-identical agreement confirms that FMBEM preserves BEM’s precision while drastically reducing computation time. This validated FMBEM served as the foundation for generating training data.

Subsequently, the CGAN model was deployed as a secondary acceleration stage. Leveraging the FMBEM-computed dataset, CGAN performed data augmentation to predict high-resolution frequency responses, further accelerating the workflow without compromising accuracy. This hybrid approach—combining FMBEM for initial acceleration and CGAN for predictive augmentation—effectively addresses the computational bottlenecks of complex acoustic simulations.

The predicted results in Fig. 17 were generated after validating the accuracy of the fast multipole algorithm. Training data were computed at 20 Hz intervals in (200–300 Hz) using the fast multipole boundary element method (FMBEM). The trained CGAN model then augmented this dataset to a finer 10 Hz resolution, and the predictions were compared with numerical solutions. The results show close agreement between the CGAN-augmented data and numerical benchmarks, with only minor deviations at specific frequencies.

The subfigures in Fig. 17 also compare the computational time required for calculating 10 data points using the boundary element method (BEM), fast multipole method (FMM), and CGAN. The FMM significantly reduces computation time compared to conventional BEM. During BEM simulations, real and imaginary components are computed simultaneously, whereas the CGAN model bypasses this complexity by learning data patterns directly. As shown in the right subfigure, CGAN generates large datasets rapidly, with computation time remaining nearly constant regardless of data volume. This efficiency confirms CGAN’s ability to accelerate computations without compromising accuracy.

The sound pressure distribution across the model surface was analyzed at four distinct frequencies, as shown in Fig. 18. At lower frequencies (100–200 Hz), the surface pressure distribution exhibits minimal variation. In contrast, at higher frequencies (400–600 Hz), significant changes occur, particularly on the washing machine’s outer surface where incident wave energy concentrates. These variations grow increasingly pronounced with rising frequency, highlighting the dynamic response of the structure under acoustic excitation.

Results of this study can help address a critical engineering challenge: understanding how vibrations generated during washing machine operation influence acoustic phenomena. These vibrations, an inevitable result of mechanical activity, often produce complex noise patterns. To investigate this interaction, vibrations were systematically applied to the washing machine model, and their effects on sound pressure distribution and acoustic performance were analyzed. The findings aim to advance theoretical insights and practical strategies for mitigating vibration-induced noise in industrial applications.

This study evaluates the CGAN model’s applicability to vibration acoustics by treating the washing machine as a rigid structure. The primary objective is to assess CGAN’s capability in predicting vibration-dependent sound pressure distributions. To quantify noise impacts under realistic conditions, two operational scenarios were simulated: Case 1 represents an observation point located at (0.5, 0, 1), while Case 2 represents an observation point at (0.5, 0, 1.8). These cases represent noise exposure at distinct heights, mimicking human ear levels in different positions. Corresponding CGAN predictions (labeled Case x-CGAN) are illustrated in Fig. 19, demonstrating the model’s ability to map vibration patterns to acoustic responses.

The results in Fig. 20 reveal distinct trends in acoustic pressure characteristics at the two observation points. Case 1 (0.5, 0, 1) exhibits markedly higher magnitudes in the real part, imaginary part, and total sound pressure compared to Case 2 (0.5, 0, 1.8). Furthermore, the acoustic pressure in Case 1 increases more sharply with frequency, suggesting that lower positions experience greater noise exposure.

The CGAN model achieves excellent agreement with reference data in predicting both complex components (real/imaginary) and total sound pressure, validating its ability to learn vibration-acoustic correlations and deliver high-fidelity predictions. This underscores CGAN’s key strength: its capacity to uncover latent data patterns irrespective of problem complexity, enabling reliable generalization across diverse scenarios.

In addition to localized noise analysis, Fig. 20 illustrates the spatial sound pressure distribution across the washing machine surface, further demonstrating the interplay between structural vibrations and acoustic responses.