Meshfree methods

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

Local RBF method for multi-dimensional partial differential equations

Abstract: In this paper, a local meshless differential quadrature collocation method is utilized to solve multi-dimensional reaction–convection–diffusion PDEs numerically In some cases, global version of the meshless method is considered as well The meshless methods approximate solution on scattered and uniform nodes in both local and global sense In the case of convection-dominated PDEs, the local meshless method is coupled with an upwind technique to avoid spurious oscillations For this purpose, a physically motivated local domain is utilized in the flow direction Both regular and irregular geometries are taken into consideration Numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains

Stabilization of finite difference methods of numerical integration

Abstract: To examine the stabilizing effects of a modification of the classical finite difference methods of numerical integration the differential equations of perturbed Keplerian motion are integrated for two examples: an artificial satellite of the Earth, and Hill's variation orbit. The modified methods remove much of the instability that is inherent to the classical methods.

The meshless Galerkin boundary node method for Stokes problems in three dimensions

Abstract: The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least-squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three-dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first-kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

Meshless methods for computational fluid dynamics

Abstract: While the generation of meshes has always posed challenges for computational scientists, the problem has become more acute in recent years. Increased computational power has enabled scientists to tackle problems of increasing size and complexity. While algorithms have seen great advances, mesh generation has lagged behind, creating a computational bottleneck. For industry and government looking to impact current and future products with simulation technology, mesh generation imposes great challenges. Many generation procedures often lack automation, requiring many man-hours, which are becoming far more expensive than computer hardware. More automated methods are less reliable for complex geometry with sharp corners, concavity, or otherwise complex features. Most mesh generation methods to date require a great deal of user expertise to obtain accurate simulation results. Since the application of computational methods to real world problems appears to be paced by mesh generation, alleviating this bottleneck potentially impacts an enormous field of problems. Meshless methods applied to computational fluid dynamics is a relatively new area of research designed to help alleviate the burden of mesh generation. Despite their recent inception, there exists no shortage of formulations and algorithms for meshless schemes in the literature. A brief survey of the field reveals varied approaches arising from diverse mathematical backgrounds applied to a wide variety of applications. All meshless schemes attempt to bypass the use of a conventional mesh entirely or in part by discretizing governing partial differential equations on scattered clouds of points. A goal of the present thesis is to develop a meshless scheme for computational fluid dynamics and evaluate its performance compared with conventional methods. The meshless schemes developed in this work compare favorably with conventional finite volume methods in terms of accuracy and efficiency for the Euler and Navier-Stokes equations. The success of these schemes may be largely attributeed their sound mathematical foundation based on a local extremum diminishing property, which has been generalized to handle local clouds of points instead of mesh-based topologies. In addition, powerful algorithms are developed to accelerate convergence for meshless schemes, which also apply to mesh based schemes in a mesh transparent manner. The convergence acceleration technique, termed "multicloud," produces schemes with convergence rates rivaling structured multigrid. However, the advantage of multicloud is that it makes no assumptions regarding mesh topology or discretization used on the finest level. Thus, multicloud is extrememly general and widely applicable. Finally, a unique application of meshless methods is demonstrated for overset grids in which a meshless method is used to seamlessly connect different types of grids. It is shown that meshless methods provide significant advantages over conventional interpolation procedures for overset grids. This application serves to highlight the practical utility of meshless schemes for computational fluid dynamics.