Meshfree methods

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

Shape Function Object Modeling for Meshfree Methods

Abstract: This article presents a computational object modeling for shape functions used in Meshfree Methods, especially those derived from the of Moving Least Square Method, such as Generalized Finite Differences, Element-Free Galerkin, Hp-Clouds and Finite Point Method, for example. The class model includes abstractions for the basis of functions, the weighting functions and the strategy of building the sparse point cloud. These abstractions allow the definition of classes of shape functions considering specific problems and methods. It should be noted that the Finite Element Method fits this class structure; the points that define the element are the cloud points in Meshfree Method. As ultimate goal, this work aims to build a framework of classes, which allows the construction of agile applications for numerical experiments of Meshfree Methods. In this work the Java language was adopted to implement the proposed design.

A Meshfree Technique for Numerical Simulation of Reaction-Diffusion Systems in Developmental Biology

Abstract: In this work, element free Galerkin (EFG) method is posed for solving nonlinear, reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. A predicator-corrector scheme is applied, to avoid directly solving of coupled nonlinear systems. The EFG method employs the moving least squares (MLS) approximation to construct shape functions. This method uses only a set of nodal points and a geometrical description of the body to discretize the governing equation. No mesh in the classical sense is needed. However a background mesh is used for integration purpose. Numerical solutions for two cases of interest, the Schnakenberg model and the Gierer-Meinhardt model, in various regions is presented to demonstrate the effects of various domain geometries on the resulting biological patterns.

Chapter 19 – The Finite Volume Particle Method: Toward a Meshless Technique for Biomedical Fluid Dynamics

Abstract: Biomedical fluid dynamics often involves large wall motions, which greatly complicate computational modeling. Meshless methods offer the potential to model moving walls in a natural manner because computational points (particles) simply move with the fluid and follow any wall or moving interface. In this chapter, one such approach, the finite volume particle method, is described. It combines some of the rigorous mathematical properties of mesh-based finite volume methods with the flexibility of meshless methods. We describe some the problems encountered in development and application of the method and their solutions. The capability to model fluid–structure interaction accurately is demonstrated with validation for a vortex-induced vibration problem. Finally, an application to an idealized heart valve is presented.

Galerkin Meshfree Methods: A Review and Mathematical Implementation Aspects

Abstract: Galerkin meshfree approaches are emerging in the field of numerical methods, which attracted the attention towards moving beyond finite element and finite difference methods. As compare to conventional mesh based finite element methods, the Galerkin meshfree methods i.e. element free Galerkin method, Local Petrov–Galerkin method, natural element method, radial point interpolation method exhibits some advantages in solution of multi-physics problem, large deformation problems, crack growth analysis due to its potential with handling variety of field behaviour. However, imposition of essential boundary conditions and nodal integration are major difficulties in Galerkin meshfree methods, this work also suggest methods for imposition of essential boundary conditions and some accurate numerical integration techniques for nodal integration. This paper gives systematic reviews and mathematical implementation aspects about Galerkin meshfree methods and its application in engineering problems i.e. structural, heat transfer, fluid dynamics and electromagnetics.