A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading

Fracture analysis of ultra-thin coating/substrate structures with interface cracks

In this study, a novel meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations. Since they form a hyperbolic system of equations, discontinuous solutions are allowed to transmit during the simulation. The generalized finite difference-split coefficient matrix method, recently proposed, is applied and improved using the flux limiter to eliminate the possible-appearing numerical oscillations. In the proposed scheme, the split-coefficient matrix method is adopted to convert the shallow water equations to the characteristic form. Then, the generalized finite difference method and the second-order Runge-Kutta method are employed for spatial and temporal discretization, respectively. The upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method. Applying the flux limiter technique, the expressions can automatically switch the proper discrete order when facing discontinuous solutions. Although the limiter function required the derivatives of different orders, the generalized finite difference method can solve these necessary expressions using the first- and second-order Tayler series. Several numerical examples are provided to demonstrate the capability of the proposed scheme, and the results are compared with other numerical schemes to show the effectiveness of the proposed generalized finite difference-flux limiter method.

A meshless generalized finite difference method for solving shallow water equations with the flux limiter technique

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems. 

/pdf/an-upwind-generalized-finite-difference-method-for-meshless-28ng1w9d.pdf

An upwind generalized finite difference method for meshless solution of two-phase porous flow equations

Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures

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Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures