Haichao Li

A unified Ritz–Equivalent Source Method (Ritz–ESM) is developed for the vibro-acoustic analysis of conical–cylindrical–spherical shells under different fluid loadings. The admissible displacement functions are constructed using Jacobi orthogonal polynomials and Fourier series, and the vibration responses under light and heavy fluid loading are obtained through the Ritz variational procedure. The surface normal velocity serves as the acoustic boundary condition. A segmented conformal equivalent source surface is constructed inside the shell, and the source strengths are recovered using the Green’s function and Tikhonov regularization to predict sound pressure, radiated sound power, and acoustic directivity. The proposed method is validated against results in the literature and FEM-BEM solutions, and the effects of truncation parameters, Jacobi parameters, source-surface geometry, and scaling factor are systematically investigated. Heavy-fluid loading shifts the natural frequencies and resonance peaks towards lower frequencies, confirming that the formulation captures the fluid added-mass effect. The reconstructed mean-square velocity and the predicted acoustic responses agree well with the structural and FEM–BEM results. The conformal source surface outperforms the axial-line and other non-conformal configurations in accuracy, numerical stability, and modeling convenience. The proposed method provides a theoretical and computational framework for predicting acoustic radiation from assembled shells of revolution and supporting parametric design.

A unified Ritz–Equivalent Source Method (Ritz–ESM) is developed for the vibro-acoustic analysis of conical–cylindrical–spherical shells under different fluid loadings. The method couples the structural energy formulation of the assembled shell with the equivalent source formulation of the radiated acoustic field. The admissible displacement functions are constructed using Jacobi orthogonal polynomials and Fourier series, and the free and forced vibration responses under light and heavy fluid loading are obtained through the Ritz variational procedure. The surface normal velocity serves as the acoustic boundary condition. A segmented conformal equivalent source surface is constructed inside the shell, and the source strengths are recovered using the Green’s function and Tikhonov regularization to predict sound pressure, radiated sound power, and acoustic directivity. The proposed method is validated against results in the literature and FEM-BEM solutions, and the effects of truncation parameters, Jacobi parameters, source-surface geometry, and scaling factor are systematically investigated. Heavy-fluid loading shifts the natural frequencies and resonance peaks towards lower frequencies, confirming that the formulation captures the fluid added-mass effect. The reconstructed mean-square velocity and the predicted acoustic responses agree well with the structural and FEM–BEM results. The conformal source surface outperforms the axial-line and other non-conformal configurations in accuracy, numerical stability, and modeling convenience. The proposed method provides a theoretical and computational framework for predicting acoustic radiation from assembled shells of revolution and supporting parametric design.

Structural health monitoring of underwater vehicles is of critical importance for ensuring their safe operation. As a key structural component, the cylindrical shell compartment plays a decisive role in the overall performance and reliability of the vehicle, with its vibration characteristics being of particular concern. However, when dealing with complex high-frequency spatial vibration fields, the finite element method (FEM) suffers from high computational cost, while conventional physics-informed neural networks (PINNs) are prone to spectral bias and gradient pathologies induced by fourth-order partial differential equations (PDEs). To address these challenges, this study proposes a feature-enhanced broadband continuous wavenumber-domain physics-informed neural network (K-PINN) for high-fidelity reconstruction of cylindrical shell vibration fields from sparse measurement data. First, a reduced-order spatial–wavenumber domain mapping model tailored for network optimization is established. By introducing second-order auxiliary physical variables, the high-order automatic differentiation barrier in deep networks is effectively eliminated. Second, a multi-scale Fourier feature embedding is innovatively integrated into the network architecture to mitigate spectral bias in broadband approximation. Numerical results demonstrate that the training time per iteration of the proposed K-PINN is reduced by two orders of magnitude compared with the standard PINN, while maintaining remarkable robustness under 20 dB Gaussian white noise and extremely sparse random sampling. Furthermore, experiments conducted on a large-scale underwater steel cylindrical shell validate the engineering applicability of the proposed framework. The overall acceleration level (OAL) reconstruction error at measurement points is strictly controlled within 0.3–0.9 dB, with typical peak errors below 3 dB.

Structural health monitoring of underwater vehicles requires accurate reconstruction of vibration responses from limited measurement data. Cylindrical shell compartments are key load-bearing components of underwater vehicles, and their broadband vibration fields often exhibit strong spatial oscillations under complex excitation. However, repeated finite element analysis is computationally expensive for broadband response evaluation, while conventional physics-informed neural networks (PINNs) may suffer from spectral bias and optimization difficulties when high-order shell governing equations are directly enforced. To address these issues, this study proposes a feature-enhanced continuous wavenumber-domain physics-informed neural network (K-PINN) for reconstructing cylindrical shell vibration responses from sparse measurements. A spatial–wavenumber mapping is established to represent the vibration field through complex k-space coefficients, and second-order auxiliary variables are introduced to avoid direct fourth-order automatic differentiation. Multi-scale Fourier features are further embedded into the network input to improve broadband representation capability. In addition, a normalized multi-component loss function, including data, governing-equation, auxiliary, boundary-condition, and sparsity constraints, is formulated to ensure stable training. Numerical results show that the proposed K-PINN reduces the per-iteration training time from 56s to 0.8s compared with a standard PINN, while maintaining accurate reconstruction under sparse sampling and 20 dB Gaussian noise. Wavenumber-domain (k-space) energy analysis further demonstrates that only 2.30% of the wavenumber coefficients are required to capture 95% of the total vibration energy at 500 Hz, supporting the physical rationality of the sparsity-promoting constraint. Underwater experiments on a large-scale steel cylindrical shell validate the engineering applicability of the proposed framework, with overall acceleration level errors within 0.3–0.9 dB and typical resonance peak errors below 3 dB.

The bandgap characteristics of periodically arranged functionally graded stepped beams are studied based on the energy variational principle in this study. The displacement field of the beam is uniformly described using an improved Fourier series. The coupling between beam segments is achieved through artificial spring technology, which simultaneously decouples the periodic boundary conditions and the shape functions. An energy method with good convergence and accuracy has been established to solve the bandgap characteristics of the beam. Through parameter research, the influence of geometric dimensions and material properties on bandgap was analyzed. The key innovation lies in introducing the simplex method to improve the beetle optimizer, optimizing the structural parameters for the first bandgap bandwidth, effectively enhancing the local search capability of the optimizer. The first bandgap bandwidth of the optimized structure has significantly increased, which has been verified through forced vibration analysis of a 10-period beam. The bandgap characteristic exhibits good stability under different boundary conditions.

The structural vibration problem in ships and marine structures is one of the core issues in this field. The typical flat plate is a basic structural unit that constitutes the main structures of the hull, such as the cabin, bow and stern, double bottom, superstructure, etc. Typical plate structures and plate frame structures are the basic structures that researchers focus on. This article briefly reviewed the development history of vibration prediction methods for typical flat plate structures and carried out work inspired by different methods and ideas. In order to provide a new tool for predicting the vibration wave energy distribution of typical flat plate structures, a new method for predicting the vibration wave energy distribution of a simple flat plate structure is proposed. This method combines the ideas of ray tracing and the finite element method, discretizes the structure into elements, and represents the transmission of vibration in the structure in the form of energy function mapping between elements and boundaries. After sufficient mapping, the steady-state solution of structural vibration energy density can be obtained. Compared with the finite element method and ray tracing method, this method has better computational efficiency; compared with statistical energy analysis, this method has better spatial resolution.