Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Four-Node Quadrilateral Element with Continuous Nodal Stress for Geometrical Nonlinear Analysis

Abstract: In this paper, the performance of a hybrid ‘FE-Meshfree’ quadrilateral element with continuous nodal stress (Quad4-CNS) is investigated for geometrical nonlinear solid mechanic problems. By combining finite element method (FEM) and meshfree method, this Quad4-CNS synergizes the individual strengths of these two methods, which leads to higher accuracy, better convergence rate, as well as high tolerance to mesh distortion. Therefore, Quad4-CNS is attractive for geometrical nonlinear solid mechanic problems where excessive distorted meshes occur. For geometrical nonlinear analysis, numerical results show that the results of Quad4-CNS element are much better than those of four-node isoparametric quadrilateral element (Quad4), and are comparable to quadratic quadrilateral element (Quad8) and other hybrid ‘FE- Meshfree’ elements.

A Modification of the Moving Least-Squares Approximation in the Element-Free Galerkin Method

Abstract: The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

A new coupled MLPG-FE method for electromagnetic field computations

Abstract: Considering an analysis of advantages and disadvantages of existing coupled finite element (FE) and meshless methods (MM), a new coupled MLPG-FE (meshless local Petrov-Galerkin-FE) method is proposed and successfully applied to electromagnetic field computations A transition region between the MM and FE domains is defined and a new ramp function is chosen In this work Instead of traditional coupled methods, this technique can ensure continuity of the displacement or potential variable and its derivatives as well as the exact implementation of Dirichlet boundary conditions The accurate numerical results for electromagnetic problems are presented

A coupled meshfree-mesh-based solution scheme on hybrid grid for flow-induced vibrations

Abstract: In this paper, a coupled meshfree-mesh-based fluid solver is employed for flow-induced vibration problems. The fluid domain comprises of a hybrid grid which is formed by generating a body conformal meshfree nodal cloud around the solid object and a static Cartesian grid which surrounds the meshfree cloud in the far field. The meshfree nodal cloud provides flexibility in dealing with solid motion by moving and morphing along with the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of a computationally efficient mesh-based method. The flow equations, in arbitrary Lagrangian–Eulerian formulation, are solved by a local radial basis function in finite difference mode on moving meshfree nodes. Conventional finite differencing is used over the static Cartesian grid for flow equations in Eulerian formulation. The equations for solid motion are solved using a classic Runge–Kutta method. Closed coupling is introduced between fluid and structural solvers by using a sub-iterative prediction–correction algorithm. In order to reduce computational overhead due to sub-iterations, only near-field flow (in the meshfree zone) is solved during the inner iterations. The full fluid domain is solved during outer (time step) iterations only when the convergence at the solid–fluid interface has already been reached. In the meshfree zone, adaptive sizing of the influence domain is introduced to maintain suitable number of neighbouring particles. The use of a hybrid grid has been found to be useful in improving the computational performance by faster computing over the Cartesian grid as well as by reducing the number of computations in the fluid domain during fluid–solid coupling. The solution scheme was tested for problems relating to flow-induced cylindrical and airfoil vibration with one and two degrees of freedom. The results are found to be in good agreement with previous work and experimental results.