In this work two particle methods are studied in the context of viscous flows. The first one is a Vortex Particle Method, called Diffused Vortex Hydrodynamics (DVH), recently developed to simulate complex viscous flows at medium and high Reynolds regimes. This method presents some similarities with the SPH model and its Lagrangian meshless nature, even if it is based on a different numerical approach. Advantages and drawbacks of the two methods have been previously studied in Colagrossi et al. [1] from a theoretical point of view and in Rossi et al. [2], where these particle methods have been tested on selected benchmarks. Further investigations are presented in this article highlighting analogies and differences between the two particle models.

A new Particle Vortex Method, called Diffused Vortex Hydrodynamics (DVH), is presented in this paper. The DVH is a meshless method characterized by the use of a regular distribution of points close to a solid surface to perform the vorticity diffusion process in the boundary layer regions. This redistribution avoids excessive clustering or rarefaction of the vortex particles providing robustness and high accuracy to the method. The generation of the regular distribution of points is performed through a packing algorithm which is embedded in the solver. The packing algorithm collocates points regularly around body of arbitrary shape allowing an exact enforcement on the solid surfaces of the no-slip boundary condition. The present method is tested and validated on different problems of increasing complexities up to flows with Reynolds number equal to 100,000 (without using any subgrid-scale turbulence model).

This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin (MLPG) method with radial basis function (RBF) kernels generated trial spaces. Local weak-form testings are done with step-functions. It is proved that subject to sufficiently many appropriate testings, solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed. Moreover, an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation. Numerical results (in double precision) give good agreement with the provided theory.

In the present work adaptation in meshless framework is proposed. The grid adaptation or mesh adaptation is quite well developed area in case of conventional grid based solvers and is popularly known as Adaptive mesh refinement (AMR). In such cases the adaptation is done by subdividing the cells or elements into finer cells or elements. In case of meshless methods there are no cells or elements but only a cloud of points. In this work we propose to achieve the meshless adaptation by locally refining the point density in the regions demanding higher resolution. This results into an adaptive enriched cloud of points. We call this method as Adaptive Cloud Refinement (ACR). The meshless solvers need connectivity information, which is a set of neighboring nodes. It is crucial part of meshless solvers. Obviously because of refining point density, the connectivity of nodes in such regions gets modified and hence has to be updated. An efficient connectivity update must exploit the fact that the node distribution would be largely unaffected except the region of adaptation. Hence connectivity updating needs to be done locally, only in these regions. In this paper we also present an extremely fast algorithm to update connectivity over adapted cloud called as ACU (Automatic Connectivity Update).

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

In this study, the element-free Galerkin (EFG) method is extended to include nonlinear behavior of cohesive cracks in 2D domains. A cohesive curved crack is modeled by using several straight-line interface elements connected to form the crack. The constitutive law of cohesive cracks is considered through the use of these interface elements. The stiffness equation of the domain is constructed by directly including, in the weak form of the global system equation, a term related to the energy dissipation along the interface elements. The constitutive law of cohesive cracks can then be considered directly and efficiently by using this energy term. The validity and efficiency of the proposed method are discussed by using problems found in the literature. The proposed method is found to be an efficient method for simulating propagation of cohesive cracks.