The Mixed LEAST SQUARES Meshfree (MDLSM) METHOD has shown its appropriate efficiency for solving Partial Differential Equations (PDEs) governing the engineering problems. The METHOD is based on the minimizing the residual functional. The residual functional is defined as a summation of the weighted residuals on the governing PDEs and the boundaries. The Moving LEAST SQUARES (MLS) is usually applied in the MDLSM METHOD for constructing the shape functions. Although the required consistency and compatibility for the approximation function is satisfied by the MLS, the METHOD loss its appropriate efficiency when the nodal points cluster too much. In the current study, the mentioned drawback is overcome using the novel approximation function called Mapped Moving LEAST SQUARES (MMLS). In this approach, the cluster of closed nodal points maps to standard nodal distribution. Then the approximation function and its derivatives compute noting the some consideration. The efficiency of suggested MMLS for overcoming the drawback of MLS is evaluated by approximating the mathematical function. The obtained results show the ability of suggested MMLS METHOD to solve the drawback. The suggested approximation function is applied in MDLSM METHOD, and used for solving the Burgers equations. Obtained results approve the efficiency of suggested METHOD.

Numerical crack modeling is an important and basic problem for researchers. Also, Finite element has good availability in crack modeling, but there are a few problems in using standard shape functions. Meshless METHODs shape functions, so DISCRETE LEAST square, which is used in this research, makes them more efficient with Finite element, especially in high-gradient problems. In this study, the cohesive crack theory leaving the usual numerical METHODs for crack behavior is investigated. In this way, doing the model conditions that do not occur crack with force out on the edges crack, and gradually become more open crack, and decrease the cohesive tension, that done zero, it means the whole crack is opening. Therefore, meshless METHODs use some techniques, such as visibility criterion and diffraction METHOD, to encounter this problem. This technique is the simplest METHOD for discontinuity modeling in meshless METHODs and has more compatibility with general domain discretization in meshless METHODs. Through engineering problems, the domain of the problem may contain nonconvex boundaries, particularly the fractured ones having discontinuous displacement fields. In such conditions, the shape functions associated with particles, whose supports intersect with the discontinuity, should be modified. One of these criteria is the visibility. In this approach, if the assumed light beam meets the discontinuity line, the shape function after the barrier will be cut. Therefore, discontinuity is applied to the geometry. In other words, the shape function of the particles, which prevents the crack or discontinuity from reaching the light beam, will be modified to amount to a zero. In this study, Cohesive Crack Theory is investigated for modeling and simulating crack behavior in DLS meshless METHOD. This METHOD gradually decreases the resistance of the cracked area to simulate splitting of the material. Finally, the high efficiency and accuracy of DLSM is given by comparing the DLSM results with experimental or FEM ones.

Water hammer” is one of the phenomena that causes damage in the pipe system and reduces their useful life. Various numerical METHODs have been used to analyze this problem. In all numerical METHODs, for calculating the variables that are the velocity and pressure values due to the sudden discontinuity of the flow and motion of the pressure wave along the pipe, the continuum environment of the problem must be discretized in some way. With calculating these aberrations before designing of the structures accurately, appropriate measures can be taken to reduce tensions caused by the water-hammer phenomenon. Background and Objectives: The conventional METHOD to numerically solve the differential equations that describe this phenomenon is the METHOD of characteristic lines. In general, in conventional METHODs where have been developed correctly, such as finite element, finite volume and finite difference, discretization of the spatial domain of the problem is done by gridding. Despite the useful use of these METHODs in many scientific fields, gridding is a costly and troublesome process, especially on problems with complex boundaries. That is the main motive for the creation of meshless METHODs. In these METHODs, the spatial domain of the problem is simply discretized by a number of points. Materials and METHODs: In the present study, for modeling classical water-hammer in a system including valve, pipe and reservoir, a collocated DISCRETE LEAST SQUARES METHOD is used. In the proposed approach implicit Crank-Nicolson METHOD for time discretization is used to provide conditions for problem solving stability. In this METHOD, the velocity and pressure values on the x-t plane are calculated directly from the previous time step simultaneously. This METHOD is quite matrix and the solution process is accomplished including several simple algebraic operations on matrices. Results: In this study, at first this numerical METHOD is described generally then governing equations are calculated and several experiments on water hammer in the form of the problem have been modeled by this METHOD, also the hydraulic analysis of problems and calculation steps for calculating accurate answers are fully described and the results are verified with exact answers and other numerical METHODs such as MOC METHOD and numerical METHOD used by “ Zielk” and the computational average error was estimated to be less than 5% by total squared error criterion. So this METHOD can be considered as a precise, simple and low-cost numerical METHOD for modeling waterhammer phenomena. Conclusion: Important properties of the Meshless numerical METHOD included no need to integrate, complete mathematical math operations and meshless space makes it one of the most accurate METHODs for numerical solution of water hammer phenomenon in the pipe system.

LEAST SQUARES is a robust and simple METHOD in function approximation. Collocated DISCRETE LEAST SQUARES (CDLS) is a meshless METHOD based on LEAST SQUARES technique. In this paper CDLS is used with a non-incremental projection METHOD for the solution of incompressible generalized Newtonian fluid flow equations in the simulation of laminar flow of power-law fluids. The scheme is used to solve two benchmark problems named lid-driven cavity flow and flow past a circular cylinder for the power-law fluids.

Meshless METHODs have been added to numerical METHODs in recent decades, and have provided a wide range of scientific, research and engineering fields. The use of Meshless METHODs is still not extent to the finite element METHODs in engineering issues, but these METHODs may now be similar to those of the time when the finite element METHOD begins to expand. In this research, a DISCRETE LEAST square meshless METHOD with collocation points CDLSM is proposed. The concepts, mathematical relations, and formulation of this METHOD are fully presented. In this simulation, collocation points are used for more efficiency and lower computing time by using LEAST SQUARES METHOD, as well as using the series instead of integrals (DISCRETE mode). Based on this METHOD, the dam failure phenomenon has been solved in different cases and its verification has been used by comparison with analytical solution with experimental data whenever it is available. Comparison of numerical results with existing analytical and experimental data shows that the METHOD has high efficiency and simulates the shock or discontinuity.

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.

In recent years, use of meshless METHODs has been extensively increased. This is probably due to the generality of their applications for the solution of continuous, as well as discontinuous, problems. Nevertheless, discretizetion of problems in many meshless METHODs, like many other numerical approaches, leads to integral equations, whose solution requires, in turn, numerical integration, definition of Gauss points, and their weight and mesh generation. Among these METHODs, however, the DISCRETE LEAST SQUARES Meshless (DLMS) METHOD has been developed, gradually, by researchers in recent years, which possesses the ability to delete integral operations from calculations of the coefficient matrix procedure.Moreover, because of its simplicity, high precision and low computational cost, this approach has been known as a real meshless METHOD. The purpose of this paper is to estimate the error of numerical solutions performed with the DISCRETE LEAST SQUARES meshless METHOD for heat conduction problems. To achieve that point, at first, the governing equations of the heat conduction problem in two dimensional space were extended, and specific boundary conditions of each problem were inserted into the formulations. Then, the DISCRETE LEAST SQUARES Meshless shape of the equations was derived for use in the proposed METHOD. Moving the LEAST SQUARES METHOD for computing the interpolation functions was undertaken. Moreover, the error estimate function was determined using the SQUARES of residuals concept. Finally, the two mentioned examples were solved. The obtained results, between the approximated proposed METHOD and the valid exact solution, which was derived from closed form analytical solutions, were compared, and the accuracy of the DISCRETE LEAST SQUARES meshless METHOD formulation was demonstrated. Furthermore, by using the LEAST SQUARES of residuals concept, error estimation was performed and error distribution or the positions of errors were obtained. By solving these examples, the power of this METHOD to solve other engineering branches, like heat conduction problems, and its high internal error diagnostic property, was illustrated.

Determination of the diffusion coefficient on the base of solution of a linear inverse problem of the parameter estimation using the LEAST-square METHOD is presented in this research. For this propose a set of temperature measurements at a single sensor location inside the heat conducting body was considered. The corresponding direct problem was then solved by the application of the heat fundamental solution.

Collocated DISCRETE LEAST SQUARES (CDLS) meshfree METHOD, is used for the numerical simulation of the Sain-Venant torsion problem and obtaining the shear stresses in the sections with irregular and complex boundaries. In this paper a matrix formulation is applied in discretizing the governing equations, which make the procedure easy to code and efficient in calculation. In the applied METHOD, a limited number of neighbor nodes are considered in the producing radial basis shape functions, which make the matrices sparse and prepare a suitable condition for sparse matrices algebra and applying available subroutine in this case. In order to validate the proposed METHOD in the analyzing stresses due to torsion, firstly an elliptic section is considered, which in this case, there is an analytical solution. In the second problem, a thin walled section is considered and solved by the theory of Saint-Venant in torsion and results are compared with the closed solution obtaining from the theory of the torsion of thin walled sections. In the third problem, torsion in a hollow elliptical section is solved. Finally, the torsion stresses in sections with more complex geometry is considered.