Meshless Methods

A function is said to have compact support if its value vanishes outside a certain distance. In other words, the function is ncmzero only within a certain region. In the finite element method, compact support ensures that the solution inside an element depends only on its nodal points. In meshless methods, the notion of compact support ensures that the number of neighboring nodes or particles affecting the solution at a certain point is finite. 

In meshless methods, the choice of the interpolation kernel is the core of the method. Various types of kernel functions are used in the literature the Gaussian kernel and spline-based kernels such as the cubic-spline, quartic, or quintic kernels are among the most frequently used kernels. 

Meshless methods belong to a class of techniques for solving boundary/initial value partial differential equations in which both geometry representation and numerical discretization are principally performed based on nodes or particles. In meshless methods, there is no inherent reliance on a particular mesh topology, meaning that no element connectivity is required. In practice, however, in many meshless methods, recourse must be taken to some kind of background meshes at least in one stage of the implementation. 

In recent years, interest in meshless methods has grown rapidly because such methods can circumvent the abovementioned difficulties in a more convenient fashion. The main advantages of meshless methods can be summarized as follows ... 

Fairly simple implementation procedure. Abundance of meshless methods, as well as... 

Based on the physical principle, meshless (particle) methods can be classified as deterministic and probabilistic. Many of the meshless methods are based on probabilistic principles. 

In reality many of meshless methods are not truly mesh free and belong to one of the following categories ... 

In 1977, Lucy [1] and Gingold and Monaghan [2] independently introduced the so-called smoothed particle hydrodynamics method which is one of the oldest meshless methods. The method was originally developed for astrophysi-cal studies such as formation of asteroids and the evolution of galaxies and has now become a standard tool in this field. In recent years, SPH has been extensively used for various fluid flow problems including both compressible and incompressible flow regimes. The method has been recently used for simulation of the generalized non-Newtonian and viscoelastic fluid flow problems. The SPH method has been also... 

In 1996, a complete volume (Vol. 139) of the journal of Computer Methods in Applied Mechanics and Engineering was devoted to meshless methods, and some of the pimieering works at the time were published. In the past 10 years, the meshless methods have constantly gained more popularity and been applied to a wider range of engineering appUcatimis. 

Today, there are books written about meshless methods [3], and several excellent review articles are also available each emphasizing on a particular class of meshless methods [4—6]. There are also several specialist groups worldwide working on various types of meshless methods. 

Meshless methods like all other computatimial methods for solving partial differential equations consist of the following general steps ... 

The main difference between the meshless methods and the mesh-based methods lies at stage three. Below, this stage is described in more detail for a number of meshless methods. 

Figure 2 presents the distribution of the weighting function around a typical node / in a two-dimensional domain along with the approximated function u x,y). Below, the approximation functions for a number of meshless methods are briefly described. 

Meshless Methods, Fig. 2 Weight function, support region, and approximated function... 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

This approximation technique is another base to construct meshless methods. In this method, the computational domain is covered by overlapping subdomains Q/ associated with each point 7. Within each subdomain I, a function ( /(x) is defined which is nonzero only in Q/ and has the following property in Q ... 

Note that the shape functions for any k in the MLS are partitions of unity. The view point used in partition of unity inspired researchers to develop several meshless methods such as the hp-clouds method and the partition of unity finite element method. 

Meshless methods are normally based on two types of discretization ... 

An important step in implementation of meshless methods is the way integrals are computed. An approach which is frequently used employs the following form ... 

The meshless methods have found their applications in various fluid mechanics and solid mechanics as well as multidisciplinary problems. They have also been successfully used for multiphysics problems such as the use of magnetohydrodynamics (MHD) to control turbulence and the study of non-Fourier heat conduction. 

The meshless methods have been successfully used in free-surface flow and moving boundaries problems. The methods have also proven to be very powerful in dealing with interfacial flow problems enabling simulation of multiphase and multi-fluid flows. The effect of surface tension has also been investigated. 

The meshless methods are particularly useful in dealing with large deformations involved in solid mechanics applications where mesh-based methods face significant difficulties. These methods have been successfully used for simulation of crack propagation under various loading conditions. 

Only during the past 6 years from the beginning of the twenty-first century, more than sixteen hundreds papers in the field of meshless methods have appeared in the literature. Despite their recent fast growth, the meshless methods still require some improvements before they can be an appropriate substitute for the standard methods such as FEM and FDM. Above all, meshless methods need a better computational speed as well as improved robustness. The computational efficiency can be increased by using parallel computing techniques as the meshless methods are naturally parallelizable. An improvement in robustness is problem dependent and needs a more thorough investigation. 

Meshless methods have high potentials to be employed in simulation of fluid-structure interactions. Complex geometries and interaction history can be defined using meshless methods requiring far less efforts. 

Coupling of the meshless methods with other standard methods such as the finite element and the finite volume methods can enhance the capabilities of both sides significantly. Such couplings can especially offer advantages in modeling problems with extreme deformations within a Lagrangian framework. This area of research can be expanded much further. 

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