A mesh-free method based on moving least squares approximation (MLS) and weak form of governing equations including two dimensional equations of motion and Maxwell’ s equation is used to analyze the free vibration of functionally graded piezoelectric material (FGPM) beams. Material properties in beam are determined using a power law distribution. Essential boundary conditions are imposed by the transformation method. The mesh-free method is verified by comparison with a finite element method (FEM) which performed for FGPM beams. Comparisons showed that this model has a good accuracy. After validation of the presented model, a parametric study was carried out to investigate the effect of mechanical and electrical boundary conditions, slenderness ratio and distribution of constituent materials on natural frequencies of FGPM beams. It is concluded that slenderness ratio has more significant effect on lower frequencies. On the other hand, higher frequencies are affected by the volume fraction power index much more than lower frequencies.

Background: The Multiwave Locked System (MLS) has been shown to reduce inflammation and enhance biostimulation. Objectives: The aim was to investigate the effect of using MLS laser in reducing pain severity in patients with chronic back pain. Methods: This randomized double-blind study was conducted on 30 patients (15 in each group) with chronic low back pain (LBP) who were referred to Imam Khomeini Hospital and diagnosed by a physical medicine specialist based on diagnostic criteria. The intervention group was treated with MLS multiple wave laser, performed 12 times, with patients receiving laser treatment twice a week. The control group was treated with exercise therapy. Pain intensity was evaluated using the Visual Analog Scale (VAS) before treatment and 6 weeks after treatment. Pain was analyzed before and after the intervention, both within and between groups. Results: Of the 30 evaluated patients, the mean and standard deviation of age (P = 0.392) and gender (P = 0.666) were not statistically significant between the two groups. The VAS value before treatment in the intervention and control groups was 7.66 ± 1.11 and 7.73 ± 1.16, respectively (P = 0.794). After treatment, the VAS values in the MLS and control groups were 5.60 ± 1.35 and 7.73 ± 1.16, respectively. Statistical evaluation showed a statistically significant difference between the two groups (P = 0.001), and the total change in VAS values was significant (P = 0.001). Conclusions: Multiwave Locked System laser can reduce the pain severity of chronic LBP, with the reduction rate in the intervention group being significantly higher than in the exercise group.

This paper proposes a novel hybrid conjugate gradient method for nonparametric statistical inference.The proposed method is a convex combination of the modified linear search (MLS) and Fletcher-Reeves (FR) methods, and it inherits the advantages of both methods. The FR method is known for its fast convergence, while the MLS method is known for its robustness to noise. The proposed method combines these advantages to achieve both fast convergence and robustness to noise. Our method is evaluated on a variety of nonparametric statistical problems, including kernel density estimation, regression, and classification. The results show that the new method outperforms the MLS and FR methods in terms of both accuracy and efficiency.

One of the main difficulties in the development of meshless methods using the Moving Least Squares approximation, such as Mixed Discrete Least Squares Meshless (MDLSM) method, is the imposition of the essential boundary conditions. In this paper the RPIM shape function, which satisfies the properties of the Kronecker delta condition, is employed in the Mixed Discrete Least Squares Meshless (MDLSM)method for solving the elasticity problems. Accordingly, two new MDLSM formulation is proposed in this article namely RPIM-based MDLSM and coupled MLS-RPIM MDLSM formulation. The essential boundary conditions can be imposed directly in both presented methods. The proposed methods are used for the solution of three benchmark elasticity problems and the results are presented and compared with the available analytical solutions and those of MLS-based MDLSM formulation. In addition, in each example different types of nodal distributions, regular and irregular configurations, are considered to test the performance of the presented methods. The numerical tests indicates higher accuracy of the suggested approaches in comparison with the MLS-based MDLSM method.

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.

The smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) are well-known and efficient mesh-less numerical methods widely used to investigate a wide range of complicated practical engineering problems. Recently, two modified Laplacian models [1, 2] have been proposed by using different efficient mathematical techniques, and the analogy between SPH and MPS methods. These two models exhibit significantly superior precision in comparison with several existing modified schemes [3-9] but still suffer from lower accuracy near calculation domain boundaries as they work with the conventional weight or interpolation functions. In this paper, the models were reformulated and further improved by replacing the weight functions with well-known moving least squares (MLS) shape functions without requiring dummy calculation nodes beyond boundaries. The proposed Laplacian models in this study could achieve very accurate results compared with the existing models [1, 2] for the solution of four different two-dimensional Poisson equations on irregular node distributions.