Selection of Sparse Sets of Influence for Meshless Finite Difference Methods

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable larg…

An Upwind Generalized Finite Difference Method (GFDM) for Meshless Analysis of Heat and Mass Transfer in Porous Media

In this paper, an upwind GFDM is developed for the coupled heat and mass transfer problems in porous media. GFDM is a meshless method that can obtain the difference schemes of spatial derivatives by using Taylor expansion in local node influence domains and the weighted least squares method. The first-order single-point upstream scheme in the FDM/FVM-based reservoir simulator is introduced to GFDM to form the upwind GFDM, based on which, a sequential coupled discrete scheme of the pressure diffusion equation and the heat convection-conduction equation is solved to obtain pressure and temperatu…

紧凑型LABFM：一种具有谱类分辨能力的无网格方法框架

Compact LABFM: a framework for meshless methods with spectral-like resolving power

本文提出一种紧凑型LABFM方法，通过优化无网格方法的分辨能力，提升复杂几何中高阶PDE模拟的准确性。

三维无网格有限差分法中改进的模板选择

Improved Stencil Selection for Meshless Finite Difference Methods in 3D

本文提出一种三维拉普拉斯算子的几何模板选择算法，改进八分圆选择策略，从不规则点集中选出高精度影响子集，并在多个STL模型的泊松方程求解中验证其性能优于有限元方法及其他选择方法。

An Upwind Generalized Finite Difference Method for Meshless Solution of Two-phase Porous Flow Equations

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 boun…

由数值微分产生的无网格无矩求积公式

Meshless moment-free quadrature formulas arising from numerical differentiation

针对Lipschitz域及其边界积分的求积权重生成问题，提出一种无需网格划分和矩计算的方法，通过求解稀疏欠定线性系统确定散乱节点权重，实验证明其在二维和三维域上鲁棒且高精度，与现有方案相比精度具竞争力。