Theoretical and Applied Mechanics Letters最新文献

The solution of fractional-order systems has been a complex problem for our research. Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive, which makes it more difficult for our solution efficiency. The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems. Therefore, this study considers how to improve the accuracy and efficiency of the solution based on traditional methods. Finally, we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model (FODS-NAR). First, we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system. Second, we compare the FODS-NAR algorithm with the famous and good reservoir computing (RC) algorithms. We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters, and the residuals of the FODS-NAR algorithm are closer to 0. Consequently, we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems. In addition, we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.

The paper studies stochastic dynamics of a two-degree-of-freedom system, where a primary linear system is connected to a nonlinear energy sink with cubic stiffness nonlinearity and viscous damping. While the primary mass is subjected to a zero-mean Gaussian white noise excitation, the main objective of this study is to maximise the efficiency of the targeted energy transfer in the system. A surrogate optimisation algorithm is proposed for this purpose and adopted for the stochastic framework. The optimisations are conducted separately for the nonlinear stiffness coefficient alone as well as for both the nonlinear stiffness and damping coefficients together. Three different optimisation cost functions, based on either energy of the system’s components or the dissipated energy, are considered. The results demonstrate some clear trends in values of the nonlinear energy sink coefficients and show the effect of different cost functions on the optimal values of the nonlinear system’s coefficients.

This work applies concepts of artificial neural networks to identify the parameters of a mathematical model based on phase fields for damage and fracture. Damage mechanics is the part of the continuum mechanics that models the effects of micro-defect formation using state variables at the macroscopic level. The equations that define the model are derived from fundamental laws of physics and provide important relationships among state variables. Simulations using the model considered in this work produce good qualitative and quantitative results, but many parameters must be adjusted to reproduce certain material behavior. The identification of model parameters is considered by solving an inverse problem that uses pseudo-experimental data to find the best values that fit the data. We apply physics informed neural network and combine some classical estimation methods to identify the material parameters that appear in the damage equation of the model. Our strategy consists of a neural network that acts as an approximating function of the damage evolution with output regularized using the residue of the differential equation. Three stages of optimization seek the best possible values for the neural network and the material parameters. The training alternates between the fitting of only the pseudo-experimental data or the total loss that includes the regularizing terms. We test the robustness of the method to noisy data and its generalization capabilities using a simple physical case for the damage model. This procedure deals better with noisy data in comparison with a more standard PDE-constrained optimization method, and it also provides good approximations of the material parameters and the evolution of damage.

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

In a global dynamic analysis, the coexisting attractors and their basins are the main tools to understand the system behavior and safety. However, both basins and attractors can be drastically influenced by uncertainties. The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors. Accordingly, analytical and numerical tools for calculation of nondeterministic global structures, namely attractors and basins, are proposed. First, based on the definition of the Perron-Frobenius, Koopman and Foias linear operators, a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases. In this context, the stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. Then, numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method. Sample results of the methodology are presented for a canonical dynamical system.

The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter. Firstly, the nonlinear system with multisource stochastic factors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposition method and the Karhunen–Loeve (K-L) decomposition theory. Secondly, the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained. At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored. Finally, the theorical results are verified by the numerical simulations.

In this letter, the effect of slip boundary on the origin of subcritical transition in two-dimensional channel flows is studied numerically and theoretically. It is shown that both the positive and the negative slip lengths will increase the critical Reynolds number of localized wave packet and hence postpone the transition. By applying a variable transformation and expanding the variables about a small slip length, it is illustrated that the slip boundary effect only exists in the second and higher order modulations of the no-slip solution, and hence explains the power law found in simulations, i.e. the relative increment of the critical Reynolds number due to the slip boundary is proportional to the square of the slip length.

In this work, we numerically study the structure of the turbulent/nonturbulent (T/NT) interface in a fully developed spatially evolving axisymmetric wake by means of direct numerical simulations. There is a continuous and contorted pure shear layer (PSL) adjacent to the outer edge of the T/NT interface. The local thickness of the PSL ${\delta }_{PSL}$ exhibits a wide range of scales (from the Kolmogorov scale to the Taylor microscale) and the conditional mean thickness $〈{\delta }_{PSL}〉{}_I/{\eta }_c\approx 6$ with ${\eta }_c$ being the centerline Kolmogorov scale is the same as the viscous superlayer. In the viscous superlayer, the pure shear motions without rotation are overwhelmingly dominant. It is also demonstrated that the physics of the turbulent sublayer is closely related to the PSL with a large thickness. Another significant finding is that the time averaged area of the rotational region $〈A_R〉$, and the pure shear region $〈A_S〉$ at different streamwise locations scale with the square of the wake-width ${b_U}^2$. This study opens an avenue for a better understanding of the structures of the T/NT interface.