Meshfree methods

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

An Efficient Solution for Volume Integral Equation Based on Meshfree Scheme

Abstract: An efficient solution for volume integral equation based on meshfree scheme is proposed. The volume current is represented by a set of shape functions in the method of moments (MoM), where the shape functions are constructed on some distributed nodes without explicitly requiring continuity conditions, so that the basis function in this letter is not reliant on the tessellation and can be used with non-conformal meshes. Numerical results demonstrate that scattering from objects can be computed efficiently with the multilevel fast multi-pole method (MLFMM).

A meshfree-based local Galerkin method with condensation of degree of freedom for elastic dynamic analysis

Abstract: Condensation technique of degree of freedom is first proposed to improve the computational efficiency of meshfree method with Galerkin weak form for elastic dynamic analysis. In the present method, scattered nodes without connectivity are divided into several subsets by cells with arbitrary shape. Local discrete equation is established over each cell by using moving Kriging interpolation, in which the nodes that located in the cell are used for approximation. Then local discrete equations can be simplified by condensation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on cells. The global dynamic system equations are obtained by assembling all local discrete equations and are solved by using the standard implicit Newmark’s time integration scheme. In the scheme of present method, the calculation of each cell is carried out by meshfree method, and local search is implemented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.

NodeLab: A MATLAB package for meshfree node-generation and adaptive refinement

Abstract: NodeLab is a simple MATLAB repository for node generation and adaptive refinement for testing, and implementing various meshfree methods for solving PDEs in arbitrary 2D domains. The core algorithm behind this package is the ‘node-placing’ approach (Fornberg & Flyer, 2015) because of its simplicity, computational speed, and the quality of the distribution. The ‘node-placing’ method has been used for creating an initial node-distribution in the boundingbox of the desired domain. A crucial decision in this context is how to represent the geometry of the domain. We compute the signed-distance field (SDF) for the domain geometry based on a priori information about the domain, which is often used in mesh-generation for finite element methods (Persson & Strang, 2004). This a priori information could be the geometry created using simple shapes, given as a function D(x, y) = 0, or some discrete set of seed nodes—later providing the flexibility to create the domain by manually digitizing the geometry. Refinement of the boundary nodes is done by formulating the problem in terms of differential equations that describe the path along the curve and interpolating through an ODE solver. The node distribution in NodeLab can be refined non-uniformly by adapting the information provided through ‘control-points’, which are an input from the user. ‘control-points’ provide spatial locations where the user needs relatively finer nodes.

Meshfree truncated hierarchical refinement for isogeometric analysis

Abstract: In this paper truncated hierarchical B-spline (THB-spline) is coupled with reproducing kernel particle method (RKPM) to blend advantages of the isogeometric analysis and meshfree methods. Since under certain conditions, the isogeometric B-spline and NURBS basis functions are exactly represented by reproducing kernel meshfree shape functions, recursive process of producing isogeometric bases can be omitted. More importantly, a seamless link between meshfree methods and isogeometric analysis can be easily defined which provide an authentic meshfree approach to refine the model locally in isogeometric analysis. This procedure can be accomplished using truncated hierarchical B-splines to construct new bases and adaptively refine them. It is also shown that the THB---RKPM method can provide efficient approximation schemes for numerical simulations and represent a promising performance in adaptive refinement of partial differential equations via isogeometric analysis. The proposed approach for adaptive locally refinement is presented in detail and its effectiveness is investigated through well-known benchmark examples.

Modeling of Grain Growth Using Voronoi Discretization and Natural Neighbor Interpolants

Abstract: Modeling of material interface evolution in grain growth of polycrystalline materials poses considerable challenges in both finite element and meshfree methods. This paper presents a Voronoi discretization in conjunction with the natural neighbor interpolants for the effective approximation of field variables with evolving interface jump conditions and topological changes of grain structures in stressed grain growth. A new algorithm for Delaunay triangulation applicable to arbitrary grain geometries is employed, and natural neighbor interpolants that provide derivative discontinuity across the material interfaces are constructed. The proposed method is applied to the modeling of grain growth processes of polycrystalline material that exhibits complex topological changes in the grain structures.