Meshfree methods

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

Panel-free boundary conditions for viscous vortex methods

Abstract: The Lagrangian vortex particle method for solving the Navier-Stokes equations is essentially a meshfree method. For the satisfaction of the boundary conditions on a submerged body, however, the need to generate a mesh (on the boundary) remains, as the only known method is based on a paneltype formulation. Here, we propose an alternative method for satisfaction of the boundary conditions which is meshfree, and based on a radial basis function collocation approach. Radial basis functions are used both to calculate the vortex sheet on the body which cancels the slip velocity, and to solve the diffusion equation which transfers the vorticity from the solid wall to the fluid domain. Proof-ofconcept calculations are presented on the impulsively-started cylinder at Reynolds numbers 200 and 1000. These results demonstrate the feasibility of the concept, and further investigations are underway to assess overall accuracy and improve computational efficiency.

On meshless and nodal-based numerical methods for forming processes

Abstract: Currently, the main tool for the simulation of forming processes is the finite element method. Unfortunately, for processes involving very large deformations, finite elements based on a Lagrangian formulation are problematic due to mesh distortion. Results become inaccurate and might even lose their physical meaning. In the 1980s, a new group of numerical methods emerged. This group is entitled meshless methods, and aims at avoiding problems related to the use of a mesh. The nodal-based approach of these methods does not restrict the relative motion of nodes by shape criteria related to elements. The goal of the research as presented in this thesis is to develop a meshless method for solving large deformations more efficiently than currently possible with finite elements. The first step in this research was to select a single meshless method for further developments. Therefore, three meshless shape functions and two numerical integration schemes were investigated in a comparative study. It can be concluded that diffuse meshless shape functions, like moving least squares and local maximum entropy approximations, are more accurate than simple linear interpolation upon a Delaunay triangle. However, the computational effort for these two diffuse functions is considerably higher. Concerning the numerical integration, a nodal integration scheme performs very well. Volumetric locking is absent and good accuracy is obtained. Therefore the linear triangle interpolation in combination with a nodal integration scheme was chosen for further development. For this combination, an extension to large deformations was presented. The new method, named Adaptive Smoothed Finite Elements (ASFEM), is based on a cloud of nodes following a Lagrangian description of motion and a triangulation algorithm that sets up the connectivity between nodes for each increment. The method was successfully tested on the simulation of a forging process and an extrusion process. No failure of the algorithm as a result of mesh distortion was encountered. Results compared accurately to reference solutions made with finite elements.

Meshfree methods for dynamic fracture

Abstract: This paper presents an overview of meshfree methods with respect to their application to dynamic fracture. Different modelling techniques for discrete cracks in meshfree methods are described. Furthermore, difficulties when dynamic fracture occurs under large deformations and fluid-structure interaction are discussed and some opportunities on how to avoid these difficulties are given. Finally, the paper closes with some future perspectives for meshfree methods

Meshless boundary particle methods for boundary integral equations and meshfree particle methods for plates

Abstract: For approximating the solution of partial dierential equations (PDE), meshless methods have been introduced to alleviate the diculties arising in mesh generation using the conventional Finite Element Method (FEM). Many meshless methods introduced lack the Kronecker delta property making them inecient in handling essential boundary conditions. Oh et al. developed several meshfree shape functions that have the Kronecker delta property. Boundary Element Methods (BEM) solve a boundary integral equation (BIE) which is equivalent to the PDE, thus reducing the dimensionality of the problem by one and the amount of computation when compared to FEM. In this dissertation, three meshless collocation based boundary element methods are introduced: meshfree reproducing polynomial boundary particle method (RPBPM), patch-wise RPBPM, and patch-wise reproducing singularity particle method (RSBPM). They are applied to the Laplace equation for convex and non-convex domains in two and three dimensions for problems with and without domain singularities. Electromagnetic wave propagation through photonic crystals is governed by Maxwell’s equations in the frequency domain. Under certain conditions, it can be shown that the wave propagation is also governed by Helmholtz equation. Patch-wise RPBPM is applied to the two dimensional Helmholtz equation and used to model electromagnetic wave propagation though lattices of photonic crystals. For thin plate problems, using the Kircho hypothesis, the three dimensional elasticity

Meshless Local Petrov-Galerkin (MLPG) Methods in Quantum Mechanics

Abstract: Purpose – The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods, particularly the one known as meshless local Petrov‐Galerkin (MLPG).Design/methodology/approach – Regarding eigenvalue problems, the authors show how to apply MLPG to the time‐independent Schrodinger equation stated in three dimensions. Through a special procedure, the numerical integration of weak forms is carried out only for internal nodes. The boundary conditions are enforced through a collocation method. The final result is a generalized eigenvalue problem, which is readily solved for the energy levels. An example of boundary value problem is described by the time‐dependent nonlinear Schrodinger equation. The weak forms are again stated only for internal nodes, whereas the same collocation scheme is employed to enforce the boundary conditions. The nonlinearity is dealt with by a predictor‐corrector scheme.Findings – Results...