Meshless approximation method of one-dimensional oscillatory Fredholm integral equations

In this findings, a numerical meshless solution algorithm for 1D oscillatory Fredholm integral equation (OFIE) is put forward. The proposed algorithm is based on Levin?s quadrature theory (LQT) incorporating multi-quadric radial basis function (MQ-RBF). The procedure involves local approach of MQ-RBF differentiation matrix. The proposed method is specially designed to handle the case when the kernel function (KF) involves stationary point(s) (SP(s)). In addition to that, the model without SP(s) is also considered. The main advantage of the meshless procedure is that it can be easily extended to multidimensional geometry. These models have several physical applications in the area of engineering and sciences. The existence of the SP(s) in such models has numerous applications in the field of scattering and acoustics etc. (see [1, 2, 4, 6-8]). The proposed meshless method is accurate and cost-effective and provides a trustworthy platform to solve OFIE(s).

MESHLESS FINITE VOLUME METHOD WITH SMOOTHING

Starting from the integral forms of the equilibrium condition and the constitutive law over the small volumes centered at the nodes, this study approximates stresses and displacements independently by means of the meshless approximation. By interpreting the meshless approximation from a new perspective, the procedure does not need to differentiate the nodal shape functions. The stresses can be approximated as accurately as the displacements, even if the shape functions for the stresses and the displacements are both taken as those simple interpolation functions such as the Shepard functions. Besides, in general no background mesh is needed. Illustrated by some elastic–plastic problems, the procedure enjoys high efficiency and excellent numerical properties.

The Development and Application of Meshless Method

Meshless method or mesh free method has many advantages. So far, there are more than ten proposed meshless methods, each has their respective advantages and disadvantages. This paper will focus on the main several meshless methods, we will make a comparison and analysis of their respective adaptation range, at the same time, we will discuss the construction method of typical meshless approximation functions, and summarize the development of the meshless method, development trend and prospects.

A STABLE LARGE DEFORMATION CORRECTED SPH METHOD BASED ON STABILIZED CONFORMING NODAL INTEGRATION AND LAGRANGIAN KERNELS

In this paper, we propose a Galerkin-based smoothed particle hydrodynamics (SPH) formulation with moving least-squares meshless approximation, applied to solid mechanics and large deformation. Our method is truly meshless and based on Lagrangian kernel formulation and stabilized nodal integration. The performance of the methodology proposed is tested through various simulations, demonstrating the attractive ability of particle methods to handle severe distortions and complex phenomena.

Dynamic Crack Problems Using Meshless Method

A Meshless Approximation Based on the Radial Basis Function (RBF) Is Developed for Analysis of Dynamic Crack Problems. A Weak Form for a Set of Governing Equations with a Unit Test Function Is Transformed into Local Integral Equations. A Completed Set of Closed Forms of the Local Boundary Integrals Are Obtained. as the Closed Forms of the Local Boundary Integrals Are Obtained, there Are No any Domain or Boundary Integrals to Be Calculated Numerically in this Approach.