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Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is m − 2 for Sobolev kernel with kernel smoothness m. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles. [ABSTRACT FROM AUTHOR]

In this paper, a physics-informed recurrent neural network (PIRNN) is proposed to solve time-dependent partial differential equations (PDEs), which devices LSTM cells to ensure the continuity of field variables in time stepping. The number of the training parameters is sharply reduced due to the parameter sharing implemented in LSTM cells so that the efficiency of PIRNN greatly improves. In order to preferably simulate the physical process and improve the accuracy of prediction, the predicted values of the current layer are employed as the input of the next layer, which exploits the idea of FDM. Thus, more information can be applied for the next prediction, and the field values at different time steps can be obtained as well. Besides, the loss value of the governing equation, the loss value of the initial condition and the loss value of the boundary condition are used to construct the loss function so that the physical law is also fully utilized. Finally, we conduct the heat conduction equation, wave equation and 2D Burgers equation to demonstrate the performance of PIRNN. Numerical experiments show that the proposed PIRNN can accurately and efficiently predict the field values at any time, in which nonuniform time steps can be used and the error accumulation is avoided. [ABSTRACT FROM AUTHOR]

In this paper, a new method named Improved Finite Integration Method (IFIM) is proposed for solving Time Fractional Diffusion Equations (TFDEs). In the IFIM, the Extended Simpson's Rule (ESR) is employed for numerical quadrature in spatial discretization. Besides, the Piecewise Quadratic Interpolation (PQI) in sense of the Hadamard finite-part integral is utilized for time discretization. Compared with the primary Finite Integration Method (FIM) with Trapezoidal rule which uses the finite difference scheme to address the time discretization, the combination of ESR and PQI in IFIM will lead to a better performance in solving TFDEs. Numerical examples are performed and compared to show the superiority of IFIM. It can also be found that the IFIM is able to obtain a higher accuracy without losing the stability and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel nonconvex and nonsmooth optimization model for low-light or inhomogeneous image segmentation which is a hybrid of Mumford–Shah energy functional and Retinex theory. The given image is decomposed into the reflectance component and the illumination component by solving Retinex-based Mumford–Shah model with L 1 − L 2 regularizer. Indeed, the existence of the L 1 − L 2 regularizer means the nonsmooth term in the model is nonconvex. Thus, it is difficult to solve the proposed model directly. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed nonconvex and nonsmooth model. We apply a novel splitting technique in our algorithm to ensure that all subproblems admit closed-form solutions. Theoretically, we prove that the sequence generated by our proposed algorithm converges to a stationary point under mild conditions. Next, once the reflectance is obtained, the K -means clustering method is utilized to complete the segmentation. We compare the proposed Retinex-based method with other state-of-the-art segmentation methods under special lighting conditions. Experimental results show that the proposed method has better performance for both gray-scale images and color images efficiently in terms of the quantitative and qualitative results. [ABSTRACT FROM AUTHOR]

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of L 2 and H 1 semi-norm errors of SNS-FEM are faster. [ABSTRACT FROM AUTHOR]

Recently, Physics-informed neural networks (PINNs) have been widely applied to solving various types of partial differential equations (PDEs) such as Poisson equation, Klein–Gordon equation, and diffusion equation. However, it is difficult to obtain higher accurate solutions, especially at the boundary due to the gradient imbalance of different loss terms for the PINN model. In this work, an adaptive learning rate residual network algorithm based on physics-informed (adaptive-PIRN) is proposed to overcome this limitation of the PINN model. In the adaptive-PIRN model, an adaptive learning rate technique is introduced to adaptively configure appropriate weights to the residual loss of the governing equation and the loss of initial/boundary conditions (I/BCs) by utilizing gradient statistics, which can alleviate gradient imbalance of different loss terms in PINN. Besides, based on the idea of ResNet, the "short connection" technique is used in adaptive-PIRN model, which can ensure that the original information is identically mapped. This structure has stronger expressive capabilities than fully connected neural networks and can avoid gradient disappearance. Finally, three different types of PDE are conducted to demonstrate predictive accuracy of our model. In addition, it is clearly observed from the results that the adaptive-PIRN can balance the gradient of loss items to a great extent, which improves the effectiveness of this network. [ABSTRACT FROM AUTHOR]

This paper presents the node-based smoothed finite element method with linear strain functions (NS-FEM-L) for solving contact problems using triangular elements. The smoothed strains are formulated by a complete order of polynomial functions and normalized with reference to the central points of smoothing domains. They are one order higher than those adopted in the finite element method (FEM) and the standard smoothed finite element method with the same triangular mesh. When using linear functions to describe strains in smoothing domains, the solutions are more accurate and stable. The contact interfaces are discretized by contact point pairs using a modified Coulomb frictional contact model. The contact problems are solved via converting into linear complementarity problems (LCPs) which can be tackled by using the Lemke method. Numerical implementations are conducted to simulate the contact behavior, including the bonding–debonding, contacting–departing and sticking–slipping. The effects of various parameters related to friction and adhesion are intensively investigated. The comparison of numerical results produced by different methods demonstrates the validity and efficiency of the NS-FEM-L for contact problems. [ABSTRACT FROM AUTHOR]