Meshless methods for CFD is an active area of research. Important developments have been made in the last few years that have contributed to the understanding of the principles that allow the approximation functions to be built and used for solving the Navier-Stokes equations.
Main Developments Documented in the Literature
The first steps towards developing meshless methods for CFD were made almost 30 years ago. The starting point for meshless research is known as the Smoothed Particle Hydrodynamics (SPH) method and dates back to the 1970s. SPH was developed by Lucy [47] and Monaghan [48] between 1977 and 1982 to model problems in Astrophysics such as explosions of stars of particle clouds. The idea behind SPH is to replace the fluid by a set of moving particles and transforming the governing partial differential equations into the kernel estimates integrals. SPH uses a pseudo-particle interpolation method to compute smooth field variables. Each pseudo-particle has mass, Lagrangian position, velocity and internal energy. Other quantities are derived by interpolation or from constitutive relations. These particles have a spatial distance (“smoothing length”), over which their properties are “smoothed” by a kernel function. Any physical quantity is then obtained by summing the relevant properties of all the particles which lie within two smoothing lengths. Although the particles are not connected in SPH, the partitioning of the domain into volume elements is difficult, especially in three dimensions.
The original ideas from SPH have had a big influence in the development of meshless methods in general. Several techniques have been developed over the years to improve the effectiveness of this method. Most of the advances were produced by incorporating powerful interpolation techniques, initially developed for data processing and surface generation. Swegle et al. [49] used dispersion analysis of the linearised equations to find the origin of the so-called ”tensile instability” and proposed an artificial viscosity to stabilize it. Dyka [50] then proposed a different stabilization method by means of stress particles. Further progress was made by Liu and Chen [51] and Liu et al. [52] by developing the Reproducing Kernel Particle Method (RKPM). The reproducing kernel in this method is similar to the SPH Lagrangian method with one major difference: the development of a correction function for boundary effects [52]. With this function, the tensile instability was eliminated. The SPH method has been successfully applied to a wide range of problems such as free surface, impact and explosion simulation, heat conduction and many other computational mechanics problems [53, 54]. Even with the many improvements that have been made throughout the years, SPH is still viewed by some as unstable and inaccurate when compared with other methods for complex fluid simulation [55], unless a large number of particles is used.
Parallel to the development of Lagrangian particle methods like SPH, Nayroles et al. [56] introduced the use of moving least square approximations in their Diffuse Ele- ment Method (DEM). The Diffuse Element Method uses moving least squares interpo- lation to replace the Finite-Element Modelling (FEM) functions, valid in one element, with a weighted minimum squares approximation, valid for a small localised domain around one point [57]. The approximation function is smoothed by introducing con- tinuous functions instead of discontinuous coefficients. The fact that these weighting functions vanish at a certain distance from the main node allows for the preservation of the local character of the approximation. It can be seen that for DEM, each of the points can be considered as a particular type of finite element, with one singular integration point, a variable number of nodes and a diffuse domain of influence.
Belytschko et al. [58] then extended the idea of least squares approximation by de- veloping a method where the spatial discretisation was made by using moving least squares and a Galerkin formulation. This scheme was called Element-Free Galerkin method (EFG) and was originally devised to solve progressive crack growth in struc- tural mechanics. The method is seen as an extension of the DEM by Nayroles, and introduced two main improvements: 1) It used an auxiliary background mesh of regular cells in order to create a structure to define the quadrature points, thus allowing for the numerical integration to be performed. 2) It was able to enforce the essential boundary conditions by using Lagrange multipliers. The method showed good accuracy, as well as convergence rates which rivalled finite element methods, even though it was compu- tationally more expensive than FE models. The EFG method has found applications in many fields such as fracture, crack and wave propagation, acoustics and fluid flow [59]. An important step towards true meshless methods was the Meshless Local Petrov- Galerkin method (MLPG), proposed by Atluri et al. [60]. The MLPG method arose from the finite element community and is based on the weak form of a given PDE [61]. MLPG incorporates the moving least squares approximations for trial and test functions to discretise the local weak form of the equations. The method is based on a Petrov- Galerkin formulation in which weight and trial functions used in the discretisation of the equations do not need to be the same. This gives the method a “local” nature in which the integral is satisfied over a local domain [61]. The MLPG approach has been used successfully to solve different problems, including work on incompressible flows [62], fracture mechanics [63], and three-dimensional elasto-statics and dynamics [63].
More recently, several methods that can be referred to as Finite Point Methods (FPMs) have been developed. They are usually based on the strong form of the PDEs. In general, FPMs are based on least squares fitting of functions to discrete points. Batina [64] proposed the use of a polynomial based approximation in conjunction with least squares to compute the derivatives of the fluid governing equations. His method provided approximations to both the Euler and Navier-Stokes equations and was suc- cessfully used to solve viscous flows about complex aircraft configurations. The first
official use of the term Finite Point Methods was provided by O˜nate [65]. He combined a weighted least square approximation of the unknowns over each local cloud with a stabilised point collocation procedure, eliminating any numerical instability. FPMs have been successfully used in several problems, including compressible inviscid and viscous flows [66, 67].
Katz and Jameson [68] developed a formal meshless scheme which compared favourably with conventional finite volume methods in terms of accuracy and efficiency for the Euler and Navier-Stokes equations. The success of their method is attributed to its local extremum diminishing property, which they generalised to handle local clouds of points instead of mesh-based schemes. The method adopts an edge-based connec- tivity to describe local points and uses Taylor series expansions with weighted least squares for the reconstruction of the gradients found in the PDEs.
Parallel Computing and Meshless Methods
Even though in recent years meshless schemes that are suitable for simulation of com- plex flows are becoming more common, most of the published work focuses on the mathematical description of the methods, without addressing the computational effi- ciency. Some researchers have implemented meshless schemes that solve the governing equations in parallel [69–72], but few have addressed the problem of parallelising the selection of local stencils. References [69, 73, 74] are among the few published works that describe the stencil selection in parallel and their method, while showing great parallel efficiency, is based on triangulation and is aimed at working on isotropic point distributions to simulate incompressible flows. To the best of the authors knowledge there has not been any published work that deals with parallel implementations for meshless schemes aimed at simulating flows over complex three-dimensional, movable geometries.