A dual-time implicit Mesh-less method is presented for unsteady compressible flow calculations. Polynomial least-square (PLS) and Taylor series least-square (TLS) procedures are used to estimate the spatial derivatives at each node and their computational efficiencies are compared. Also, the effect of the neighbor stencil selection on the accuracy of the method is investigated. As a new approach, different neighboring stencils are used for the highly stretched point distribution inside the boundary layer region and the inviscid isotropic point distribution outside this area. The unsteady flows over stationary and moving objects at subsonic and transonic flow conditions are solved. Results indicate the computational efficiency of the method in comparison with the alternative approaches. The convergence histories of the flow solution show that the PLS method is computationally faster than TLS method. In addition, the eight point neighboring stencil inside the viscous region is more efficient than other choices.

An implicit Mesh-less method is developed for calculation of compressible flows around three dimensional complex geometries. The algorithm is applied directly to the differential form of the governing equations using least-square formulation. A dual-time implicit time discretization scheme is developed, and the computational efficiency is enhanced by adopting accelerating techniques, such as local time stepping, residual smoothing and enthalpy damping. Two different artificial dissipation techniques are employed for stability preservation and it is shown that the scalar one is more efficient in terms of accuracy and computational time. The capabilities of the method are demonstrated by flow computations around different geometries under subsonic and transonic flow conditions. Results are presented which indicate good agreement with experimental and other reliable numerical data. The method is shown to reduce computational time by about 50% compared with the alternative explicit method.

A formulation of the Element Free Galerkin (EFG), one of the Mesh-less methods, is developed for solving coupled problems and its validity for application to soil-water problems is examined through numerical analysis. The numerical approach is constructed to solve two governing partial differential equations of equilibrium and the continuity of pore water, simultaneously. Spatial variables in a weak form, the displacement increment and excess pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used to create the discretization of the time domain. Implementation of essential boundary conditions is based on penalty method. Examples are studied and the obtained results are compared with closed-form or finite element method solutions to demonstrate the capability of the developed model. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well, both soil deformation and the variation of pore water pressure, over time.

In this paper, a study is performed on the effect of irregularity of domain discretization on the performance of the CDLSM method for the solution of convection-dominated problems. The method is based on minimizing a least squares functional of the residuals of the governing differential equations and its boundary conditions over a set of collocation points. Four convection-dominated benchmark examples are solved using CDLSM method on three different sets of nodal distribution with different levels of irregularity and the results are presented. These experiments show that CDLSM method is capable of producing stable and accurate results for hyperbolic problems with shocked or high gradient solutions even on highly irregular Mesh of nodes. Mesh-less methods as alternative numerical approaches to eliminate the well-known drawbacks of Mesh-based methods have attracted much attention in the past decade due to their edibility and their potentiality in negating the need for the human-labor intensive process of constructing geometric Meshes in a domain. Exploiting this ability, however, requires that the method could solve the problem under consideration on unstructured distribution of nodes. This is particularly important when a refinement strategy is to be used to improve the performances of these methods.