Meshless generalized finite difference method for two- and three-dimensional transient elastodynamic analysis

This paper presents an arbitrary order numerical approach for achieving highly accurate solutions in transient heat conduction problems. In this framework, the arbitrary order Krylov deferred correction method (KDC) is first utilized to discretize the temporal direction of the problems, ensuring high-accuracy results by selecting an adequate number of Gaussian nodes. During each time iteration, a boundary value problem in terms of the inhomogeneous modified Helmholtz equation is generated by utilizing the KDC methodology. Subsequently, the arbitrary order GFDM is introduced to solve the boundary value problem, allowing for flexible selection of the Taylor series expansion order to improve spatial accuracy. Consequently, the present numerical framework is of arbitrary order of accuracy and very stable through integrating advantages of the arbitrary order KDC and the arbitrary order GFDM. The developed numerical framework is ultimately verified for its characteristic of high accuracy and stability through two numerical examples in two-dimensions and three numerical examples in three-dimensions. The results highlight the significant improvement in numerical accuracy achieved by the present arbitrary-order numerical approach for long-time simulations. 

An arbitrary order numerical framework for transient heat conduction problems

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods. 

A stable numerical framework for long-time dynamic crack analysis

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

This paper firstly establishes the discrete-time lattice networks for nonlocal stochastic competitive neural networks with reaction diffusions and fuzzy logic by employing a mix techniques of finite difference to space variables and Mittag-Leffler time Euler difference to time variable. The proposed networks consider both the effects of spatial diffusion and fuzzy logic, whereas most of the existing literatures focus only on discrete-time networks without spatial diffusion. Firstly, the existence of a unique ω-anti-periodic in distribution to the networks is addressed by employing Banach contractive mapping principle and the theory of stochastic calculus. Secondly, global exponential convergence in mean-square sense to the networks is discussed on the basis of constant variation formulas for sequences. Finally, an illustrative example is used to show the feasible of the works in the current paper with the help of MATLAB Toolbox. The work in this paper is pioneering in this regard and it has created a certain research foundations for future studies in this area.

Anti-periodic motion and mean-square exponential convergence of nonlocal discrete-time stochastic competitive lattice neural networks with fuzzy logic

This article presents a meshless method to solve three-dimensional elliptic interface problem. The method is based on the generalized finite difference method, which expresses the derivatives of unknown variables by linear combinations of nearby function values. The proposed method turns the interface problem into some boundary value sub-problems coupled by the interface conditions. This conversion leads to an important feature, that is, the proposed method is not sensitive to the jump coefficients or the interface's geometry. It can handle different complex interfaces by only changing the level set function of the interface in the process. Several numerical examples with sufficient complexity verify the accuracy and stability of the method. For some given examples, the method is more accurate than the classical immersed finite element method. 

A meshless method based on the generalized finite difference method for three-dimensional elliptic interface problems

In this paper, we propose the neural finite element method (NFEM) with the two-part structure to obtain a highly accurate result for generalized Black-Scholes-Merton equation arising in the financial market. The two-layer rectified linear unit in the first part can be regarded as the adaptive subdivision of the domain into several elements, and the nonlinear activation function in the second is similar to the numerical solution of each element. The shallow and deep neural networks are connected to fit high-frequency and low-frequency components, and the deep residual module is proposed to improve the approximate ability of the NFEM. We apply the NFEM to solve three generalized BSM equations with different boundary conditions, including the European call option and the American put option. Through comparison with other methods by numerical experiments, our work shows that NFEM provides an exciting new tool for this model. 

Numerical solving for generalized Black-Scholes-Merton model with neural finite element method

We developed a visual teaching platform that can calculate the magnetic field of magnetic core inductance in real time. The platform adopts the combination of two theories of finite element calculation and neural network technology. It can enhance students’ understanding and application of the basic knowledge of electromagnetic fields. First, the finite element method was used to calculate the magnetic field of the magnetic core inductance, and the magnetic field data set under different input parameters was obtained. On this basis, the neural network method was used to learn the data set and train the corresponding model. Then the trained neural network model was used to calculate the magnetic core inductance magnetic field. After comparing with the finite element calculation results, we found that the calculation results of the neural network model combined with the finite element were in good agreement with the finite element calculation results. Compared with the finite element calculation method, the calculation speed of the magnetic field distribution calculated by the neural network was faster. Taking the calculation of the magnetic core inductance magnetic field as an example, the calculation time was shortened by about 170 times. Finally, we built a magnetic field visualization teaching platform based on MATLAB. The example magnetic field was quickly predicted by the neural network, saving computing time and effectively improving the teaching of electromagnetic field theory.

/pdf/magnetic-field-visualization-teaching-based-on-fusion-method-1kvuvj9h.pdf

Magnetic Field Visualization Teaching Based on Fusion Method of Finite Element and Neural Network

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A meshless method based on the generalized finite difference method for three-dimensional elliptic interface problems

Numerical solving for generalized Black-Scholes-Merton model with neural finite element method

Magnetic Field Visualization Teaching Based on Fusion Method of Finite Element and Neural Network