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.

Management and exploitation in mines require a continuous and relatively smooth surface of the mineral grades. While assessing the various mineral elements, the scattered exploratory cavities are irregularly excavated. Producing a continuous surface from measured data requires interpolation methods. Several factors, including the characteristics of the data, affect the efficiency of the interpolation methods. For this reason, the efficiency of different methods in various cases is inconsistence, and choosing the appropriate interpolation method is also challenging. Interpolation methods can be categorized into two groups of mesh-based and meshless methods. Despite the efficiency and capabilities of meshless methods, they have a fundamental shortcoming due to the fixed size of the support domain. On the one hand, the distribution of exploratory cavities in mines is usually irregular, and in some areas, it is very dense, and in others, it is very sparse. On the other hand, the grade values of minerals at the surface of the region can be very variable with high changes. Conventional interpolation methods do not have sufficient efficiency and flexibility in confronting these two aforementioned issues. In this study, a precise, reliable, and flexible method is developed for interpolation of minerals through integrating the MOVING LEAST squares and recursive LEAST squares methods. In the proposed method for crack detection, the residuals statistical test of LEAST squares computations is used. In this method, for the central point, a continuity threshold (non-continuity) is determined based on the standard deviation of field values, so that points with crack are revealed and removed from the calculation of the value of the central point. Moreover, the size of the support domain is determined dynamically based on the recursive property of the method. In this method, an individual radius for the support domain is assigned to each central point according to the values and distributions of the surrounding field points. The dynamic size of the support domain allows a precise and reliable estimation of polynomial coefficients and the values of the central points. The efficiency of the proposed method is evaluated by applying it to simulated data as well as comparing it with the results of conventional interpolation methods on real mineral data. The results of the simulation data indicate the ability of the proposed method to reveal the non-continuity and fractures of surfaces with determining the dynamics size of the support domain based on the data structure. To compare the results of the proposed method with conventional interpolation methods including LPI, IDW, Kriging, and RBF, the root mean square error (RMSE), mean and median of errors are used. In this way, in addition to the overall accuracy of each method, the distribution of errors is also determined. The RMSE, mean and median errors of the proposed method, using the 10-fold cross-validation method for chromium (Cr), are 28. 020, 0. 2. 201 and 2. 874, respectively, and for iron (Fe) are 1. 074, 0. 017 and 0. 094, respectively. Comparison of these results with conventional interpolation methods indicates the efficiency of the proposed method for both groups of high concentration and significant changes in the values and low concentration and almost uniform level of values. The results indicate the ability of the proposed method in detecting the jumps and non-continuity in the support domain and removal of some field points within the dynamic process, lead to a significant increase in the efficiency of the method compared to conventional methods.

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.

Conceptual design optimization of spacecraft systems is a complex and multidisciplinary process. In this case evaluation of the objective functions relies heavily on running iterative simulation models and analysis codes between various subsystems (such as structures, payload, electrical power supply, attitude determination and control, communication, command and data handling). The conventional sequential optimization approaches to such a complex design problem is time consuming and does not guarantee to achieve the best compromise among the various competing coupled subsystems, and may even lead to non-optimal design. In addition, the design search space can be multi-modal, non-convex with multiple local minima and hence it is time consuming or difficult to rapidly evaluate trade-offs between various subsystems (disciplines). To address these issues, in this paper an efficient surrogate (response surface) model-based multidisciplinary spacecraft systems design optimization technique with discrete and continuous design variables is presented. The methodology is based on the utilization of genetic algorithms (GA) for both system level and discipline level as an optimizer. Surrogate-modeling as an efficient tool is also used to decrease computational cost in discipline (subsystem) level within a collaborative optimization (CO) framework. Results obtained in this study show that the method introduced in this paper provides an effective way of improving computational efficiency of a complex space system design such as conceptual design optimization of a spacecraft.

The MOVING LEAST Square (MLS) interpolation method is proposed for approximation of adaptive fuzzy controller parameters for two degrees of freedom suspension system and each one has two inputs, one output with twenty-five linguistic fuzzy IF-THEN rules. Fuzzy systems are designed by using five Gaussian membership functions for each input, product inference engine, singleton fuzzifier and center average defuzzifier. The constructed fuzzy systems is composed with adaptation rules. For this purpose, Lyapunove approach is implemented for stability of the adaptation rules. The Gravity Search Algorithm (GSA) is implemented for achieve the optimum controller parameters. The relative displacement between sprung mass and tire and the body acceleration are two objective functions used in the optimization algorithm. Since, choose the suitable controller coefficients are important and when the parameter of the system change, Optimum coefficients of the controller will also change. In order to solve this obstacle, the MLS predictive model is purposed that is interpolation method based on a radius of the neighborhood, a basis function and a weight function for points of interest. Finally online model is implemented on the two degrees of freedom suspension system and results compared with the offline optimal systems.

In terms of observational data, there are some problems in the standard Big Bang cosmological model. Inflation era, early accelerated phase of the evolution of the universe, can successfully solve these problems. The inflation epoch can be explained by scalar inflaton field. The evolution of this field is presented by a non-linear differential equation. This equation is considered in FLRW model. In FLRW model, we consider the universe as the warped product of real line with a three dimensional homogeneous and isotropic manifold  which could have positive, negative or zero curvature. The main aim of this paper is the numerical solution of the inflation evolution differential equations using of a meshless discrete Galerkin method. The method reduces the solution of these types of differential equations to the solution of Volterra integral equations of the second kind. Therefore, we solve these integral equations using MOVING LEAST squares method. Finally, a numerical example is included to show the validity and efficiency of the new technique.

We present a new and simple direct approach based on generalized MOVING LEAST squares (GMLS) for computing the first derivatives of the functions defined on the sphere. The novel method utilizes a Householder transformation (reflection) and a projection onto the tangent plane to compute the first derivatives at the original point on the sphere. The main benefit of this algorithm is that there is no need to use the spherical harmonics for constructing the approximation of the first derivatives. An example of the approximation has been tested to show the ability of the developed method. Moreover, this method has been applied to solve the transport equation in one example.

In this research, a linear combination of MOVING LEAST square (MLS) and local radial basis functions (LRBFs) is considered within the framework of the meshless method to solve the two-dimensional hyperbolic telegraph equation. Besides, the differential quadrature method (DQM) is employed to discretize temporal derivatives. Furthermore, a control parameter is introduced and optimized to achieve minimum errors via an experimental approach. Illustrative examples are provided to demonstrate the applicability and efficiency of the method. The results prove the superiority of this method over using MLS and LRBF individually.

In this paper, the interpolating MOVING LEAST-squares (IMLS) method is discussed. The interpolating MOVING LEAST square methodology is an e, ective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for numerical solution of Volterra{Fredholm integral equations, and , nally some examples are given to show the accuracy and applicability of the method.