Meshfree methods

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

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Abstract: Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.

Least‐squares meshfree method for incompressible Navier–Stokes problems

Abstract: A least-squares meshfree method based on the first-order velocity-pressure-vorticity formulation for two-dimensional incompressible Navier-Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least-squares method. Equal-order moving least-squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix-free element-by-element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier-Stokes problem and the flow over a square obstacle for time-dependent Navier-Stokes problem are investigated for the presented least-squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated

A Meshfree Method for Solving Cardiac Electrical Propagation

Abstract: We present a novel numerical scheme to accurately and efficiently simulate the spatiotemporal electrical propagation for three dimensional heart model. A meshfree particle representation of myocardial volume is first developed, upon which the electrical propagation can be obtained using the element-free Galerkin (EFG) method for the FitzHugh-Nagumo model. This method is based on a sufficient amount of sampling nodes of the three-dimensional myocardial volume, but without the needs to construct the often expensive and complicated mesh structure between these nodes. Compared to the traditional finite element method, this new approach provides a more efficient numerical method to model the effects of the myocardial geometrical complexity and material inhomogeneity/anisotropicness. Experiments on synthetic and real heart geometries with uniform and nonuniform diffuse materials are presented. Related implementation issues are also discussed