Applied Mathematics and Mechanics-English Edition最新文献

The parametric surrogate models for partial differential equations (PDEs) are a necessary component for many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool to generate these surrogates when parametric fields are present. CNNs are commonly trained on labeled data based on one-to-one sets of parameter-input and PDE-output fields. Recently, residual-based deep convolutional physics-informed neural network (DCPINN) solvers for parametric PDEs have been proposed to build surrogates without the need for labeled data. These allow for the generation of surrogates without an expensive offline-phase. In this work, we present an alternative formulation termed deep convolutional Ritz method (DCRM) as a parametric PDE solver. The approach is based on the minimization of energy functionals, which lowers the order of the differential operators compared to residual-based methods. Based on studies involving the Poisson equation with a spatially parameterized source term and boundary conditions, we find that CNNs trained on labeled data outperform DCPINNs in convergence speed and generalization abilities. The surrogates generated from the DCRM, however, converge significantly faster than their DCPINN counterparts, and prove to generalize faster and better than the surrogates obtained from both CNNs trained on labeled data and DCPINNs. This hints that the DCRM could make PDE solution surrogates trained without labeled data possibly.

Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in the MD displacement data sets. Inspired by the weighted essentially non-oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as the piecewise smooth solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization-based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the first phase, we identify the optimal nonlocal kernel function from the data sets without material damage to capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with MD. We illustrate the efficacy of the proposed approach with several numerical tests for the dynamic crack propagation problem in a single-layer graphene. Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable of modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training.

Physics-informed neural networks (PINNs) are proved methods that are effective in solving some strongly nonlinear partial differential equations (PDEs), e.g., Navier-Stokes equations, with a small amount of boundary or interior data. However, the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported. The present paper proposes an artificial viscosity (AV)-based PINN for solving the forward and inverse flow problems. Specifically, the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics (CFD) to stabilize the simulation of flow at high Reynolds numbers. The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re = 1 000 and the inverse problem derived from two-dimensional film boiling. The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.

This paper develops a deep learning tool based on neural processes (NPs) called the Peri-Net-Pro, to predict the crack patterns in a moving disk and classifies them according to the classification modes with quantified uncertainties. In particular, image classification and regression studies are conducted by means of convolutional neural networks (CNNs) and NPs. First, the amount and quality of the data are enhanced by using peridynamics to theoretically compensate for the problems of the finite element method (FEM) in generating crack pattern images. Second, case studies are conducted with the prototype microelastic brittle (PMB), linear peridynamic solid (LPS), and viscoelastic solid (VES) models obtained by using the peridynamic theory. The case studies are performed to classify the images by using CNNs and determine the suitability of the PMB, LBS, and VES models. Finally, a regression analysis is performed on the crack pattern images with NPs to predict the crack patterns. The regression analysis results confirm that the variance decreases when the number of epochs increases by using the NPs. The training results gradually improve, and the variance ranges decrease to less than 0.035. The main finding of this study is that the NPs enable accurate predictions, even with missing or insufficient training data. The results demonstrate that if the context points are set to the 10th, 100th, 300th, and 784th, the training information is deliberately omitted for the context points of the 10th, 100th, and 300th, and the predictions are different when the context points are significantly lower. However, the comparison of the results of the 100th and 784th context points shows that the predicted results are similar because of the Gaussian processes in the NPs. Therefore, if the NPs are employed for training, the missing information of the training data can be supplemented to predict the results.

A high-order gas kinetic flux solver (GKFS) is presented for simulating inviscid compressible flows. The weighted essentially non-oscillatory (WENO) scheme on a uniform mesh in the finite volume formulation is combined with the circular function-based GKFS (C-GKFS) to capture more details of the flow fields with fewer grids. Different from most of the current GKFSs, which are constructed based on the Maxwellian distribution function or its equivalent form, the C-GKFS simplifies the Maxwellian distribution function into the circular function, which ensures that the Euler or Navier-Stokes equations can be recovered correctly. This improves the efficiency of the GKFS and reduces its complexity to facilitate the practical application of engineering. Several benchmark cases are simulated, and good agreement can be obtained in comparison with the references, which demonstrates that the high-order C-GKFS can achieve the desired accuracy.

The influence of weights is usually ignored in the study of nonlinear vibrations of plates. In this paper, the effect of structure weights on the nonlinear vibration of a composite circular plate with a rigid body is presented. The nonlinear governing equations are derived from the generalized Hamilton’s principle and the von Kármán plate theory. The equilibrium configurations due to weights are determined and validated by the finite element method (FEM). A nonlinear model for the vibration around the equilibrium configuration is established. Moreover, the natural frequencies and amplitude-frequency responses of harmonically forced vibrations are calculated. The study shows that the structure weights introduce additional linear and quadratic nonlinear terms into the dynamical model. This leads to interesting phenomena. For example, considering weights increases the natural frequency. Furthermore, when the influence of weights is considered, the vibration response of the plate becomes asymmetrical.

The escalation of zeta potential by the influence of wall slip for the electrokinetically modulated flow through a microchannel motivates to consider the impact of hydrodynamic slippage upon the zeta or surface potential. The reported study undergoes an analytical exploration of the pulsatile electroosmosis and shear-actuated flow characteristics of a fluid with a Newtonian model through a microchannel with parallel plates by invoking the reliance of a zeta or surface potential on slippage. The linearized Poisson-Boltzmann and momentum equations are solved analytically to obtain the explicit expression of the electrical potential induced in the electrical double layer (EDL), the flow velocity field, and the volumetric flow rate for an extensive span of parameters. The velocity field proximal to the microchannel wall is observed to enhance by an apparent zeta potential, and is further escalated for a thinner EDL and an oscillating electric field with a higher amplitude. However, near the core region of the microchannel, the flow velocity becomes invariant with the EDL thickness. The result shows that the lower wall velocity contributes to the flow velocity along with the electroosmotic body force and the impact of the velocity of the wall underneath diminishes proximal to the upper wall. Moreover, the volumetric flow rate increases when the thickness of the EDL decreases, owing to the influence of the wall slip. However, for thinner EDLs and medium and higher oscillating Reynolds numbers, the volumetric flow rate varies non-monotonously, correlative to the slip-free and slip cases.

The ingenious hierarchical structure of enamel composed of rods and protein produces excellent fracture resistance. However, the fracture resistance mechanism in the inner enamel is unknown. The micromechanical models of enamel are constructed to numerically analyze the mechanical behaviors of the inner enamel with different decussation angles and different decussation planes. The results show that the manner of crack propagation in the inner enamel, including crack bridging, crack deflection, and crack bifurcation, is determined by both the rod decussation angle and the decussation plane. In the case of the strong decussation plane, the fracture strength and the required energy dissipation with the decussation angles of 15° and 30° are much higher than those without decussation, demonstrating that decussation is an important mechanism in improving the fracture resistance of enamel. The maximum tensile stress of enamel with the decussation angle of 15° is slightly higher than that of enamel with the decussation angle of 30°, illustrating that an optimal decussation angle exists which balances the strength and toughness. The synergetic mechanism of the decussation angle and the decussation plane on the crack propagation provides a new design hint for bionic composites.

In this study, a coupling model of fluid-conveying pipes made of functionally graded materials (FGMs) with NiTiNOL-steel (NiTi-ST) for vibration absorption is investigated. The vibration responses of the FGM fluid-conveying pipe with NiTi-ST are studied by the Galerkin truncation method (GTM) and harmonic balance method (HBM). The harmonic balance solutions and the numerical results are consistent. Also, the linearized stability of the structure is determined. The effects of the structure parameters on the absorption performance are also studied. The results show that the NiTi-ST is an effective means of vibration absorption. Furthermore, in studying the effect of the NiTi-ST, a closed detached response (CDR) is first observed. It is noteworthy that the CDR may dramatically change the vibration amplitude and that the parameters of the NiTi-ST may determine the emergence or disappearance of the CDR. This vibration absorption device can be extended to offer more general vibration control in engineering applications.

The similitude theory helps to understand the physical behaviors of large structures through scaled models. Several papers have studied the similitude of shock issues. However, the dynamic similitude for shock responses of coupled structures is rarely incorporated in open studies. In this paper, scaling laws are derived for the shock responses and spectra of coupled structures. In the presented scaling laws, the geometric distortion and energy loss are considered. The ability of the proposed scaling laws is demonstrated in the simulation and experimental cases. In both cases, the similitude prediction for the prototype’s time-domain waveform and spectrum is conducted with the scaled model and scaling laws. The simulation and experimental cases indicate that the predicted shock responses and spectra agree well with those of the prototype, which verifies the proposed scaling laws for predicting shock responses.