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A meshless method namely, discrete least square method (DLSM), is presented in the paper for the solution of free surface seepage problem. In this method computational domain is discredited by some nodes and then the set of simultaneous equations are built using moving least square (MLS) shape functions and least square technique. The proposed method does not need any background mesh therefore it is a truly meshless method. Several numerical two dimensional examples of Poisson partial differential equations (PDEs) are presented to illustrate the performance of the present DLSM. And finally a free surface seepage problem in a porous media is solved and results are presented.

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A meshless approach, collocation discrete least square (CDLS) method, is extended in this paper, for solving

elasticity problems. In the present CDLS method, the problem domain is discretized by distributed field nodes. The field

nodes are used to construct the trial functions. The moving least-squares interpolant is employed to construct the trial

functions. Some collocation points that are independent of the field nodes are used to form the total residuals of the

problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of the

residuals for the collocation points. The final stiffness matrix is symmetric and therefore can be solved via efficient

solvers. The boundary conditions are easily enforced by the penalty method. The present method does not require any

mesh so it is a truly meshless method. Numerical examples are studied in detail, which show that the present method

is stable and possesses good accuracy, high convergence rate and high efficiency.

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A least squares based meshfree method is used in the numerical simulation of a turbulent flow. The proposed approach is integral free, vectorized and enjoying sparse positive definite matrices. Here the standard k-ε model is employed to model the turbulent flow. A matrix formulation is illustrated that simply can be extended for other turbulence models. Two bench mark problems are solved and results are compared with the literature.

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A numerical meshless method is proposed to investigate shallow water equations. Because of The numerical solution of the pure convection equations represents a very important issue in many problems, an Element Free Galerkin (EFG) method is used for solving these equations, and its implementation is described. In this method there is no need to nodal connectivity and just uses nodal data which may be the same as those used in the Finite Element Methods (FEMs) and a description of the domain boundary geometry are necessary. The essential boundary condition is enforced by the penalty method, and the Moving Least Squares (MLS) approximation is used for the interpolation scheme. The numerical efficiency of the proposed method is demonstrated by solving several benchmark examples. Sensitivity analysis on parameters of the EFG method is carried out and results are presented.