Applied Mathematics and Mechanics-English Edition最新文献

This paper proposes a method to amplify the performance of a flexural-wave-generation system by utilizing the energy-localization characteristics of a phononic crystal (PnC) with a piezoelectric defect and an analytical approach that accelerates the predictions of such wave-generation performance. The proposed analytical model is based on the Euler-Bernoulli beam theory. The proposed analytical approach, inspired by the transfer matrix and S-parameter methods, is used to perform band-structure and time-harmonic analyses. A comparison of the results of the proposed approach with those of the finite element method validates the high predictive capability and time efficiency of the proposed model. A case study is explored; the results demonstrate an almost ten-fold amplification of the velocity amplitudes of flexural waves leaving at a defect-band frequency, compared with a system without the PnC. Moreover, design guidelines for piezoelectric-defect-introduced PnCs are provided by analyzing the changes in wave-generation performance that arise depending on the defect location.

This paper proposes a quasi-zero stiffness (QZS) isolator composed of a curved beam (as spider foot) and a linear spring (as spider muscle) inspired by the precise capturing ability of spiders in vibrating environments. The curved beam is simplified as an inclined horizontal spring, and a static analysis is carried out to explore the effects of different structural parameters on the stiffness performance of the QZS isolator. The finite element simulation analysis verifies that the QZS isolator can significantly reduce the first-order natural frequency under the load in the QZS region. The harmonic balance method (HBM) is used to explore the effects of the excitation amplitude, damping ratio, and stiffness coefficient on the system’s amplitude-frequency response and transmissibility performance, and the accuracy of the analytical results is verified by the fourth-order Runge-Kutta integral method (RK-4). The experimental data of the QZS isolator prototype are fitted to a nine-degree polynomial, and the RK-4 can theoretically predict the experimental results. The experimental results show that the QZS isolator has a lower initial isolation frequency and a wider isolation frequency bandwidth than the equivalent linear isolator. The frequency sweep test of prototypes with different harmonic excitation amplitudes shows that the initial isolation frequency of the QZS isolator is 3 Hz, and it can isolate 90% of the excitation signal at 7 Hz. The proposed biomimetic spider-like QZS isolator has high application prospects and can provide a reference for optimizing low-frequency or ultra-low-frequency isolators.

A novel vibration absorber is designed to suppress vibrations in fluid-conveying pipes subject to varying fluid speeds. The proposed absorber combines the fundamental principles of nonlinear energy sinks (NESs) and nonlinear energy harvesters (NEHs). The governing equation is derived, and a second-order discrete system is used to assess the performance of the developed device. The results demonstrate that the proposed absorber achieves significantly enhanced energy dissipation efficiency, reaching up to 95%, over a wider frequency range. Additionally, it successfully harvests additional electric energy. This research establishes a promising avenue for the development of new nonlinear devices aimed at suppressing fluid-conveying pipe vibrations across a broad frequency spectrum.

Although most pipes are restrained by retaining clips in aircrafts, the influence of the clip parameters on the vibration of the fluid-conveying pipe has not been revealed. By considering the clip width, a new dynamic model of a fluid-conveying pipe restrained by an intermediate clip is established in this paper. To demonstrate the necessity of the proposed model, a half pipe model is established by modeling the clip as one end. By comparing the two models, it is found that the half pipe model overestimates the critical velocity and may estimate the dynamical behavior of the pipe incorrectly. In addition, with the increase in the clip stiffness, the conversion processes of the first two modes of the pipe are shown. Furthermore, by ignoring the width of the clip, the effect of the flow velocity on the accuracy of a concentrated restraint clip model is presented. When the flow velocity is close to the critical velocity, the accuracy of the concentrated restraint clip model significantly reduces, especially when the width of the clip is large. In general, the contribution of this paper is to establish a dynamic model of the fluid-conveying pipe which can describe the influence of the clip parameters, and to demonstrate the necessity of this model.

The Schwarz primitive triply periodic minimal surface (P-type TPMS) lattice structures are widely used. However, these lattice structures have weak load-bearing capacity compared with other cellular structures. In this paper, an adaptive enhancement design method based on the non-uniform stress distribution in structures with uniform thickness is proposed to design the P-type TPMS lattice structures with higher mechanical properties. Two types of structures are designed by adjusting the adaptive thickness distribution in the TPMS. One keeps the same relative density, and the other keeps the same of non-enhanced region thickness. Compared with the uniform lattice structure, the elastic modulus for the structure with the same relative density increases by more than 17%, and the yield strength increases by more than 10.2%. Three kinds of TPMS lattice structures are fabricated by laser powder bed fusion (L-PBF) with 316L stainless steel to verify the proposed enhanced design. The manufacture-induced geometric deviation between the as-design and as-printed models is measured by micro X-ray computed tomography (µ-CT) scans. The quasi-static compression experimental results of P-type TPMS lattice structures show that the reinforced structures have stronger elastic moduli, ultimate strengths, and energy absorption capabilities than the homogeneous P-TPMS lattice structure.

Physics-informed deep learning has recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the over-parameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In paper “MENG, X., YANG, L., MAO, Z., FERRANDIS, J. D., and KARNIADAKIS, G. E. Learning functional priors and posteriors from data and physics. Journal of Computational Physics, 457, 111073 (2022)”, a Bayesian framework based on the generative adversarial networks (GANs) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in “MENG, X., YANG, L., MAO, Z., FERRANDIS, J. D., and KARNIADAKIS, G. E. Learning functional priors and posteriors from data and physics. Journal of Computational Physics, 457, 111073 (2022)” has two stages: (i) prior learning, and (ii) posterior estimation. At the first stage, the GANs are utilized to learn a functional prior either from a prescribed function distribution, e.g., the Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference (VI), which naturally enables the mini-batch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional (100D) Darcy problem, are conducted to demonstrate that the NFs with full-/mini-batch training are able to achieve similar accuracy as the “gold rule” HMC. Moreover, the mini-batch training of NF makes it a promising tool for quantifying uncertainty in solving the high-dimensional partial differential equation (PDE) problems with big data.

We propose a self-supervising learning framework for finding the dominant eigenfunction-eigenvalue pairs of linear and self-adjoint operators. We represent target eigenfunctions with coordinate-based neural networks and employ the Fourier positional encodings to enable the approximation of high-frequency modes. We formulate a self-supervised training objective for spectral learning and propose a novel regularization mechanism to ensure that the network finds the exact eigenfunctions instead of a space spanned by the eigenfunctions. Furthermore, we investigate the effect of weight normalization as a mechanism to alleviate the risk of recovering linear dependent modes, allowing us to accurately recover a large number of eigenpairs. The effectiveness of our methods is demonstrated across a collection of representative benchmarks including both local and non-local diffusion operators, as well as high-dimensional time-series data from a video sequence. Our results indicate that the present algorithm can outperform competing approaches in terms of both approximation accuracy and computational cost.

We consider solving the forward and inverse partial differential equations (PDEs) which have sharp solutions with physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose the adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only-based ASM denoted by ASM I. In this approach, we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains, then we add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have the largest absolute values of the residual in this sub-domain as new residual points. We further develop a second type of ASM (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I, and we add new residual points which have not only large residuals but also large gradients. To demonstrate the effectiveness of the present methods, we use both ASM I and ASM II to solve a number of PDEs, including the Burger equation, the compressible Euler equation, the Poisson equation over an L-shape domain as well as the high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II, and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency, and stability compared with the ASM I algorithm. This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution. Furthermore, we also employ the similar adaptive sampling technique for the data points of boundary conditions (BCs) if the sharpness of the solution is near the boundary. The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency, stability, and accuracy.

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.