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.

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.

The main purpose of this article is to increase the efficiency of the LEAST SQUARES METHOD in numerical solution of ill-posed functional and physical equations. Determining the LEAST SQUARES of a given function in an arbitrary set is often an ill-posed problem. In this article, by defining artificial constraint and using Lagrange multipliers METHOD, the attempt is to turn n-dimensional LEAST SQUARES problems into (n-1) ones, in a way that the condition number of the corresponding system with(n-1) -dimensional problem will be low. At first, the new METHOD is introduced for2 and 3-term basis, then the presented METHOD is generalized for n-term basis. Finally, the numerical solution of some ill-posed problems like Fredholm integral equations of the first kind and singularly perturbed linear Fredholm integral equations of the second kind are approximated by chain LEAST SQUARES METHOD. Numerical comparisons indicate that the chain LEAST SQUARES METHOD yields accurate and stable approximations in many cases.

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.

The concise review systematically summarises the state-of-the-art variants of Moving LEAST SQUARES (MLS) METHOD. MLS METHOD is a mathematical tool which could render cogent support in data interpolation, shape construction and formulation of meshfree schemes, particularly due to its flexibility to form complex arithmetic equation. However, the conventional MLS METHOD is suffering to deal with discontinuity of field variables. Varied strategies of overcoming such shortfall are discussed in current work. Although numerous MLS variants were proposed since the introduction of MLS METHOD in numerical/statistical analysis, there is no technical review made on how the METHODs evolve. The current review is structured according to major strategies on how to improvise MLS METHOD: the modification of weight function, the manipulation of discrete norms, the inclusion of iterative feature for residuals minimising and integration of these strategies for more robust computation. A wide range of advanced MLS variants have been compiled, summarised, and reappraised according to its underlying principle of improvement. In addition, inherent limitation of MLS METHOD and its possible strategy of improvement is discussed too in this article. The current work could render valuable reference to implement and develop advanced MLS schemes, whenever complexity of the specific scientific problems arose.

Summary Global and regional geomagnetic field models give the components of the geomagnetic field as functions of place and time. Most of these models utilize polynomials or Fourier series to map the input variables to the geomagnetic field values. The only temporal variation in these models is the long term secular variation. However, there is an increasing need amongst certain users for the models that can provide shorter term temporal variations, such as the geomagnetic daily variation. In this research, we have constructed an empirical model of the quiet daily geomagnetic field variation based on functional fitting...

This paper introduces a novel METHOD for solving the generalized total LEAST SQUARES problem, an extension of the total LEAST SQUARES problem. The generalized total LEAST SQUARES problem emerges when solving overdetermined linear systems with the multiple right-hand sides $\mathbf{AX} \thickapprox \mathbf{B}$, where both the observation matrix $\mathbf{B}$ and the data matrix $\mathbf{A}$ contain errors. Our approach involves extending the Taylor series expansion to reformulate the generalized total LEAST SQUARES problem into a linear problem, allowing us to employ the tensor form of the generalized LEAST SQUARES algorithm for efficient computation. This technique streamlines the computational process and enhances solution accuracy. For a more detailed survey, we compare the proposed METHOD for solving the generalized total LEAST SQUARES problem with one of the matrix format METHODs for the associated total LEAST SQUARES problem. Empirical results show that our METHOD significantly improves computational efficiency and solution precision. Additionally, we demonstrate its practical application in the context of image blurring.

In this paper, the laminar incompressible flow equations are solved by an upwind LEAST-SQUARES meshless METHOD. Due to the difficulties in generating quality meshes, particularly in complex geometries, a meshless METHOD is increasingly used as a new numerical tool. The meshless METHODs only use clouds of nodes to influence the domain of every node. Thus, they do not require the nodes to be connected to form a mesh and decrease the difficulty of meshing, particularly around complex geometries. In the literature, it has been shown that the generation of points in a domain by the advancing front technique is an order of magnitude faster than the unstructured mesh for a 3D configuration. The Navier–Stokes solver is based on the artificial compressibility approach and the numerical METHODology is based on the higher-order characteristic-based (CB) discretization. The main objective of this research is to use the CB scheme in order to prevent instabilities. Using this inherent upwind technique for estimating convection variables at the mid-point, no artificial viscosity is required at high Reynolds number. The Taylor LEAST-SQUARES METHOD was used for the calculation of spatial derivatives with normalized Gaussian weight functions. An explicit four stage Runge-Kutta scheme with modified coefficients was used for the discretized equations. To accelerate convergence, local time stepping was used in any explicit iteration for steady state test cases and the residual smoothing techniques were used to converge acceleration. The capabilities of the developed 2D incompressible Navier-Stokes code with the proposed meshless METHOD were demonstrated by flow computations in a lid-driven cavity at four Reynolds numbers. The obtained results using the new proposed scheme indicated a good agreement with the standard benchmark solutions in the literature. It was found that using the third order accuracy for the proposed METHOD could be more efficient than its second order accuracy discretization in terms of computational time.

This paper presents the analysis of tapered slot antenna by using the METHOD of LEAST SQUARES (MLS). For this purpose first the conductor and dielectric regions are divided into suitable subsections. Then we assign the proper basis functions for their unknown currents. Therefore, the problem is changed to determining these currents: surface currents on conductor region and the polarization volume current in dielectric region of antenna. For solving the problem using MLS, we define the error function, which is based on a combination of two equations that satisfy the antenna electromagnetic conditions and then minimize it. So by computing these currents, we determine the far field radiation of the antenna. At the end of the paper, E-plane and H-plane patterns of several antennas and compare them with results of other papers that use the moment METHOD.