This paper proposes an efficient deep encoder–decoder ConvLSTM neural network for solving time-dependent differential partial equations. The encoder is composed of convolution layers, and the decoder is composed of transposed convolution layers, where the image is converted from low frequency to high frequency, and then back to low frequency in order to extract the spatial features of the partial differential equation. To accurately capture temporal information of time-dependent partial differential equations, ConvLSTM is configured in the encoder and decoder to solve the long-term evolution of dynamic systems. Furthermore, skip connections are added between the convolutional layer and the corresponding transposed convolutional layer, which not only overcomes the problem of gradient disappearance but also ensures that the underlying features are learned, enhancing the accuracy and efficiency of our network. Finally, the proposed network has been used to solve heat equations, wave equations, 2D Burgers equations, and high-order equations, and the results show that the present neural network has much better performance in terms of accuracy and efficiency.

This paper aims to develop an efficient and accurate numerical method for solving the third-order nonlinear Whitham–Broer–Kaup (WBK) shallow water model. To achieve this objective, we applied a cubic B-spline-based finite element method, which offers higher-order continuity and precise geometric representation. The study of this model is important due to its wide applications in fluid dynamics, plasma physics, and nonlinear optics. We established the existence and uniqueness of the numerical solutions and performed a convergence analysis for the semi-discrete Galerkin scheme. The system of ordinary differential equations produced by the application of this method is solved by the Crank–Nicolson method. To validate the current approach, three test problems are analyzed. The errors are measured by using the [Formula: see text] and [Formula: see text] norms. Numerical results demonstrate that the method provides highly accurate approximations, with close agreement between numerical and exact solutions, confirming the efficiency, stability, and reliability of the cubic B-spline finite element method for modeling complex nonlinear wave phenomena. In addition, we compared the present method with several published methods. The results show that the proposed method provides a better solution in terms of absolute error and computational efficiency.

An intramedullary (IM) femur nail is an orthopedic implant for treating femur fractures, designed to bear body weight and mechanical stresses, reducing the load on fractured bone segments and allowing early patient mobilization. This study examines the biomechanical four-point bending test (ASTM F1264) of three-dimensional (3D)-printed fiber-reinforced polyether ether ketone (PEEK) IM nails, using both experimental methods and numerical simulation. Finite element analysis predicted the behavior of fiber-reinforced PEEK IM nails in a fractured femur model. Load-bearing capacity, bending strength, and stiffness were compared between carbon fiber (CF)- and glass fiber (GF)-reinforced PEEK IM nails. Experimental load versus displacement data showed strong correlation with simulation results, validating the finite element model. The CF PEEK nail withstood a force of 1,783.14[Formula: see text]N and showed 39.34% higher bending loads and a stiffness of 168.22[Formula: see text]N/mm, outperforming the GF PEEK nail’s 1,279.97[Formula: see text]N force and 120.75[Formula: see text]N/mm stiffness. Stress analysis indicated CF PEEK sustained higher stress (116.6[Formula: see text]MPa) compared to GF PEEK (83.68[Formula: see text]MPa), establishing CF PEEK as the superior material. The biomechanical tests demonstrated CF PEEK’s superior bending resistance and toughness over GF PEEK. The combined experimental and computational approach provided a comprehensive understanding of the mechanical behavior and stress responses of the implants, validating CF PEEK as a superior material and highlighting the potential of 3D printing with fiber-reinforced PEEK for orthopedic implants. The integration of experimental validation and computational prediction is essential for optimizing implant design and improving patient outcomes in orthopedic treatments.

This study investigates the free vibrational analysis of cylindrical shells made of Porous Functionally Graded Materials (PFGMs) under arbitrary boundary conditions. To present more general formulations, it is assumed that the cylinder is partially bounded by an elastic foundation, characterized by either constant or variable stiffness. Utilizing the First-order Shear Deformation Theory (FSDT) and applying the Hamilton’s principle, the governing differential equations of motion have been obtained. An analytical solution i.e., Rayleigh–Ritz method combined with Lagrange multipliers has been used to solve these equations. It is worth noting that the inclusion of a partially elastic foundation results in the coupled equations between the cylinder’s length and angular directions, which is addressed through the series-solution. After validating the results for different boundary conditions and porosity types, we proceed to explore the influence of porosity function, boundary conditions, geometric parameters, and elastic foundation on the free vibrational characteristics. This comprehensive analysis provides insights into the behavior of PFGMs cylindrical shells, aiding in the design and optimization of structures utilizing these materials.

In the nonparametric modeling of hydropneumatic suspension, the effective terms of nonlinear parameters are often difficult to determine, which further leads to the difficulty of obtaining the structural stiffness and damping characteristics. To solve this problem, this paper develops a novel method based on the polynomial structure selection technique to determine the effective terms of the nonparametric model of the hydropneumatic suspension, and then the stiffness and damping will be obtained. First, the hydropneumatic suspension bench test is carried out to obtain the output force, the excitation speed and the excitation displacement under different working conditions. Then, the stiffness and damping characteristics of the hydropneumatic suspension are theoretically analyzed. Finally, through the analysis results and the polynomial structure selection technique, the simplest structure of nonparametric mathematical model for the hydropneumatic suspension model is obtained, and later the stiffness and damping of the hydropneumatic suspension are identified. Comparison between the test output force of the hydropneumatic suspension and the output force of the simplified model shows that the simplified model can be used to replace the real model, and eventually verifies that the damping and stiffness characteristics of the hydropneumatic suspension are only related to the excitation displacement and the excitation speed, and the two ones do not interfere with each other.

A systematic study is presented toward optimizing various variants of mass matrices for the interior-node rotation-free quintic Euler–Bernoulli beam element. This Hermite type beam element consists of four nodes, where the two end nodes contain both deflectional and rotational degrees of freedom (DOFs), but the two interior nodes are solely associated with the deflections. By introducing an adjustable nodal distance parameter between the end and adjacent nodes, a nodal-distance adjustable mass matrix is devised. Accordingly, both diagonal and block-diagonal mass matrices of this four-node quintic Hermite beam element are developed with respect to different nodal distance parameters, which are determined through optimizing the corresponding frequency accuracy in the context of nodal integration. It is shown that the best performance of the diagonal mass matrix is a sub-optimal frequency convergence, while the optimal convergence can be achieved by the block-diagonal mass matrix. Subsequently, a higher-order mass matrix featured by two extra orders of frequency accuracy is formulated via optimally blending the consistent and block-diagonal mass matrices. Moreover, regarding the diagonal mass matrices, it is proved that the interior-node rotation-free four-node element yields superior frequency accuracy in comparison with the conventional three-node element where all nodes have the deflectional as well as rotational DOFs. An excellent agreement is observed between the theoretical and numerical results.

This paper studies the cubic B-spline finite element approximation of an optimal control problem governed by the thin film epitaxy equation. The state and co-state variables are approximated by cubic B-spline finite elements, while the control variable is approximated by piecewise constant functions. The backward Euler scheme is employed for the time discretization. We derive some a priori error estimates for both the control and state variables. Numerical experiments are performed to verify the theoretical results.

The vector nozzle will inevitably produce uncertain joint clearance during the manufacturing, assembly and working processes. The kinematic accuracy and dynamic response of the vector nozzle mechanism will be affected by the uncertainty of joint clearance. The joint clearance uncertainty causes the strong nonlinearity of the vector nozzle mechanism. This paper proposes an analysis method for the kinematic accuracy and dynamic response of the rigid-flexible coupling vector nozzle mechanism with uncertain joint clearance based on the deep neural network. The uncertainty analysis process based on the deep neural network model and Latin hypercubic sampling method is given to illustrate the relationship between the joint clearance and the dynamic response of the nozzle mechanism. The results show that the uncertainty of joint clearance should be considered for the rigid-flexible coupling vector nozzle mechanism in order to accurately assess its performance and to provide a reference for the design, analysis and assessment of the vector nozzle mechanism.

This paper presents a numerical method combining the Differential Quadrature Method (DQM) with Newton linearization (DQNL) for solving nonlinear Lane–Emden equations. DQM discretizes the domain, yielding a nonlinear algebraic system solved via Newton’s method. The proposed approach is applied to several test problems with various nonlinearities. Stability under noise perturbations, execution time, and accuracy are thoroughly analyzed. Numerical results show that DQNL achieves high accuracy with few discretization points and iterations, outperforming existing methods in both precision and efficiency.