Meshfree methods

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

Implementation of Material Interface Conditions in the Radial Point Interpolation Meshless Method

Abstract: We propose a systematic approach to accurate imposition of material interface conditions for the meshless radial point interpolation method (RPIM). A new set of equations for updating fields at interfaces is derived. A piecewise polynomial and a modified radial basis function are proposed and applied to account for field discontinuities near the interfaces. In contrast to the previous work, the proposed approach incorporates material properties and reduces geometrical dependence of the interpolation function. Numerical results show that the proposed approach significantly improves the simulation accuracy of the RPIM at a cost of a small overhead in computational time.

On the Use of Meshless Methods in Acoustic Simulations

Abstract: In this paper, we are investigating the potential and limits of a meshless Lagrangian technique, called Smoothed Particle Hydrodynamics (SPH), as a method for acoustic simulations. Currently the most common techniques for acoustic simulations draw on mesh-based methods such as the Boundary Element Method (BEM), Finite Differences Method (FD) and Finite Element Method (FEM). Though many improvements have been made to each class of methods during the last few years, they still have their weaknesses. Difficulties arise as soon as inhomogeneous media, moving boundaries or aeroacoustic effects are involved. These problems are either particularly hard to describe or cannot be simulated with some of these mesh-based methods. The investigation of SPH for modeling sound propagation is carried out in order to assess its potential in relation to the limitations associated with the existing simulation methods listed above. Simple computational experiments will be carried out for the verification of the new approach and applications on the problems listed above will be discussed.Copyright © 2009 by ASME

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method*

Abstract: In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional (2D) elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions

Abstract: In this paper, the meshless local radial point interpolation (MLRPI) method is formulated to the generalized one-dimensional linear telegraph and heat diffusion equation with non-local boundary conditions. The MLRPI method is categorized under meshless methods in which any background integration cells are not required, so that all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique based on the radial point interpolation is adopted to construct shape functions, also called basis functions, using the radial basis functions. These shape functions have delta function property in the frame work of interpolation, therefore they convince us to impose boundary conditions directly. The time derivatives are approximated by the finite difference time- -stepping method. We also apply Simpson’s integration rule to treat the non-local boundary conditions. Convergency and stability of the MLRPI method are clarified by surveying some numerical experiments.

A radial basis function method for solving PDE-constrained optimization problems

Abstract: In this article, we apply the theory of meshfree methods to the problem of PDE-constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and also discuss the Neumann boundary control problem, both involving Poisson's equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modification to guarantee invertibility. We implement these methods using Matlab, and produce numerical results to demonstrate the methods' capability. We also comment on the methods' effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended.