Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems

An enriched finite element approach with appropriate enrichment function is proposed for the transient analyses of wave propagations in nonhomogeneous media. In present method, the traditional linear nodal interpolation functions for low order finite elements are enriched by the augmented enrichment function. For the purpose of improving the computation accuracy and reducing the numerical dispersion error, the polynomial bases and the trigonometric functions are utilized to form the enrichment functions. In addition, a simple and direct approach is introduced with the aim to tackle the possible linear dependence of the system equations. The numerical results show that the accuracy of the obtained numerical results can be markedly improved by the proposed method in comparison with the standard finite elements. What's more, it can be found that the calculated solutions from the proposed method with the appropriate temporal discretization scheme are able to monotonically converge to the exact solution in transient wave analysis. In consequence, it is possible to consistently increase the solution accuracy by directly decreasing temporal discretization step size. This valuable and important property enable the present approach to be particularly suitable to tackle the wave propagations in nonhomogeneous media. 

Transient analyses of wave propagations in nonhomogeneous media employing the novel finite element method with the appropriate enrichment function

This paper introduces a hybrid numerical technique aimed at effectively addressing two-dimensional (2D) non-linear transient heat conduction problems, featuring temperature-dependent thermal conductivity. This scheme involves integrating the Krylov deferred correction (KDC) method in the temporal discretization and the generalized finite difference method (GFDM) in the spatial discretization. The core of the KDC method introduces a new unknown variable in the form of the first-order time derivative of temperature, leading to a non-linear equation after temporal discretization. Subsequently, the corresponding non-linear equation is solved in the spatial domain using the GFDM, supported by the Jacobian-free Newton-Krylov (JFNK) solver and fourth-order Taylor series expansion. The accuracy and stability of the hybrid approach are verified through two numerical experiments, demonstrating the strong performance of the present algorithm known as the KDC-GFDM for the simulations of the interested problems in long-time intervals. 

A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity

Analysis of a superconvergent recursive moving least squares approximation

This work presents a novel simulation approach to couple the meshfree radial point interpolation method (RPIM) with the implicit direct time integration method for the transient analysis of wave propagation dynamics in non-homogeneous media. In this approach, the RPIM is adopted for the discretization of the overall space domain, while the discretization of the time domain is completed by employing the efficient Bathe time stepping scheme. The dispersion analysis demonstrates that, in wave analysis, the amount of numerical dispersion error resulting from the discretization in the space domain can be suppressed at a very low level when the employed nodal support domain of the interpolation function is adequately large. Meanwhile, it is also mathematically shown that the amount of numerical error resulting from the time domain discretization is actually a monotonically decreasing function of the non-dimensional time domain discretization interval. Consequently, the present simulation approach is capable of effectively handling the transient analysis of wave propagation dynamics in non-homogeneous media, and the disparate waves with different speeds can be solved concurrently with very high computation accuracy. This numerical feature makes the present simulation approach more suitable for complicated wave analysis than the traditional finite element approach because the waves with disparate speeds always cannot be concurrently solved accurately. Several numerical tests are given to check the performance of the present simulation approach for the analysis of wave propagation dynamics in non-homogeneous media.

/pdf/the-meshfree-radial-point-interpolation-method-rpim-for-wave-jnscrf3p.pdf

The Meshfree Radial Point Interpolation Method (RPIM) for Wave Propagation Dynamics in Non-Homogeneous Media

The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results. 

Theoretical analysis of the generalized finite difference method

This paper proposes and analyzes a stabilized element-free Galerkin (EFG) method for meshless Galerkin analysis of the generalized Stokes problem. The Nitsche-type weak form of the generalized Stokes problem is derived by adopting Nitsche's technique to deal with the lack of interpolation property of meshless shape functions. By introducing the residual-based stabilization technique to establish a stabilized Nitsche-type EFG weak form, the proposed stabilized EFG method not only allows equal-order approximation spaces for velocity and pressure, but also applies to the generalized Stokes problem with small viscosity and large reaction coefficient. Both the inf-sup stability and the error estimation of the stabilized EFG method are analyzed theoretically. Some numerical results are provided to demonstrate the efficiency of the method. 

Meshless Galerkin analysis of the generalized Stokes problem

Smoothed gradients of meshless shape functions have been widely used in meshless methods to enhance the performance of solving partial differential equations. In this paper, properties and error estimation of the moving least squares approximation with smoothed gradients are analyzed theoretically. Numerical results are also presented to evince the extra reproduction properties and superconvergence of smoothed gradients and confirm the theoretical analysis. 

Analysis of the moving least squares approximation with smoothed gradients

Analysis of an element-free Galerkin method for the nonlinear Schrödinger equation

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