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The problems of shear flexible beams are analyzed by the MLPG method based on a locking‐free weak formulation. In order for the weak formulation to be locking‐free, the numerical characteristics of the variational functional for a shear flexible beam, in the thin beam limit, are discussed. Based on these discussions a locking‐free local symmetric weak form is derived by changing the set of two dependent variables in governing equations from that of transverse displacement and total rotation to the set of transverse displacement and transverse shear strain. For the interpolation of the chosen set of dependent variables (i.e. transverse displacement and transverse shear strain) in the locking‐free local symmetric weak form, the recently proposed generalized moving least squares (GMLS) interpolation scheme is utilized, in order to introduce the derivative of the transverse displacement as an additional nodal degree of freedom, independent of the nodal transverse displacement. Through numerical examples, convergence tests are performed. To identify the locking‐free nature of the proposed method, problems of shear flexible beams in the thick beam limit and in the thin beam limit are analyzed, and the numerical results are compared with analytical solutions. The potential of using the truly meshless local Petrov‐Galerkin (MLPG) method is established as a new paradigm in totally locking‐free computational analyses of shear flexible plates and shells.

This paper aims to introduce a new numerical approach based on the local weak form and the Petrov–Galerkin idea to numerically simulation of a predator–prey system with two-species, two chemicals and an additional chemotactic influence.

In the first proceeding, the space derivatives are discretized by using the direct meshless local Petrov–Galerkin method. This generates a nonlinear algebraic system of equations. The mentioned system is solved by using the Broyden’s method which this technique is not related to compute the Jacobian matrix.

This current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.

The proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

The study aims to highlight the behaviour of one-dimensional and two-dimensional fin models under the natural room conditions, considering the different values of dimensionless Biot number (Bi). The effect of convection and radiation on the heat transfer process has also been demonstrated using the meshless local Petrov–Galerkin (MLPG) approach.

It is true that MLPG method is time-consuming and expensive in terms of man-hours, as it is in the developing stage, but with the advent of computationally fast new-generation computers, there is a big possibility of the development of MLPG software, which will not only reduce the computational time and cost but also enhance the accuracy and precision in the results. Bi values of 0.01 and 0.10 have been taken for the experimental investigation of one-dimensional and two-dimensional rectangular fin models. The numerical simulation results obtained by the analytical method, benchmark numerical method and the MLPG method for both the models have been compared with that of the experimental investigation results for validation and found to be in good agreement. Performance of the fin has also been demonstrated.

The experimental and numerical investigations have been conducted for one-dimensional and two-dimensional linear and nonlinear fin models of rectangular shape. MLPG is used as a potential numerical method. Effect of radiation is also, implemented successfully. Results are found to be in good agreement with analytical solution, when one-dimensional steady problem is solved; however, two-dimensional results obtained by the MLPG method are compared with that of the finite element method and found that the proposed method is as accurate as the established method. It is also found that for higher Bi, the one-dimensional model is not appropriate, as it does not demonstrate the appreciated error; hence, a two-dimensional model is required to predict the performance of a fin. Radiative fin illustrates more heat transfer than the pure convective fin. The performance parameters show that as the Bi increases, the performance of fin decreases because of high thermal resistance.

Though, best of the efforts have been put to showcase the behaviour of one-dimensional and two-dimensional fins under nonlinear conditions, at different Bi values, yet lot more is to be demonstrated. Nonlinearity, in the present paper, is exhibited by using the thermal and material properties as the function of temperature, but can be further demonstrated with their dependency on the area. Additionally, this paper can be made more elaborative by extending the research for transient problems, with different fin profiles. Natural convection model is adopted in the present study but it can also be studied by using forced convection model.

Fins are the most commonly used medium to enhance heat transfer from a hot primary surface. Heat transfer in its natural condition is nonlinear and hence been demonstrated. The outcome is practically viable, as it is applicable at large to the broad areas like automobile, aerospace and electronic and electrical devices.

As per the literature survey, lot of work has been done on fins using different numerical methods; but to the best of authors’ knowledge, this study is first in the area of nonlinear heat transfer of fins using dimensionless Bi by the truly meshfree MLPG method.

The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods, particularly the one known as meshless local Petrov‐Galerkin (MLPG).

Regarding eigenvalue problems, the authors show how to apply MLPG to the time‐independent Schrödinger equation stated in three dimensions. Through a special procedure, the numerical integration of weak forms is carried out only for internal nodes. The boundary conditions are enforced through a collocation method. The final result is a generalized eigenvalue problem, which is readily solved for the energy levels. An example of boundary value problem is described by the time‐dependent nonlinear Schrödinger equation. The weak forms are again stated only for internal nodes, whereas the same collocation scheme is employed to enforce the boundary conditions. The nonlinearity is dealt with by a predictor‐corrector scheme.

Results show that the combination of MLPG and a collocation scheme works very well. The numerical results are compared to those provided by analytical solutions, exhibiting good agreement.

The flexibility of MLPG is made explicit. There are different ways to obtain the weak forms, and the boundary conditions can be enforced through a number of ways, the collocation scheme being just one of them. The shape functions used to approximate the solution can incorporate modifications that reflect some feature of the problem, like periodic boundary conditions. The value of this work resides in the fact that problems from other areas of electromagnetism can be attacked by the very same ideas herein described.

1) The OE-MLPG penalty meshfree method is developed to solve cracked structure.(2) Smartening the numerical meshfree method by combining the particle swarm optimization (PSO) optimization algorithms and Voronoi computational geometric algorithm. (3). Selection of base functions, finding optimal penalty factor and distribution of appropriate nodal points to the accuracy of calculation in the meshless local Petrov–Galekrin (MLPG) meshless method.

Using appropriate shape functions and distribution of nodal points in local domains and sub-domains and choosing an approximation or interpolation method has an effective role in the application of meshless methods for the analysis of computational fracture mechanics problems, especially problems with geometric discontinuity and cracks. In this research, computational geometry technique, based on the Voronoi diagram (VD) and Delaunay triangulation and PSO algorithm, are used to distribute nodal points in the sub-domain of analysis (crack line and around it on the crack plane).

By doing this process, the problems caused by too closeness of nodal points in computationally sensitive areas that exist in general methods of nodal point distribution are also solved. Comparing the effect of the number of sentences of basic functions and their order in the definition of shape functions, performing the mono-objective PSO algorithm to find the penalty factor, the coefficient, convergence, arrangement of nodal points during the three stages of VD implementation and the accuracy of the answers found indicates, the efficiency of V-E-MLPG method with Ns = 7 and ß = 0.0037–0.0075 to estimation of 3D-stress intensity factors (3D-SIFs) in computational fracture mechanics.

The present manuscript is a continuation of the studies (Ref. [33]) carried out by the authors, about; feasibility assessment, improvement and solution of challenges, introduction of more capacities and capabilities of the numerical MLPG method have been used. In order to validate the modeling and accuracy of calculations, the results have been compared with the findings of reference article [34] and [35].

The work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to solve steady-state heat conduction in 2-D as well as in 3-D domains.

The restarted version of the GMRES solver (with and without preconditioner) is applied to solve an asymmetric system of equations, arising due to the interpolating MLPG formulation. Its performance is compared with the biconjugate gradient stabilized (BiCGSTAB) solver on the basis of computation time and convergence behaviour. Jacobi and successive over-relaxation (SOR) methods are used as the preconditioners in both the solvers.

The results show that the GMRES solver outperforms the BiCGSTAB solver in terms of smoothness of convergence behaviour, while performs slightly better than the BiCGSTAB method in terms of Central processing Unit (CPU) time.

MLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods such as GMRES and BiCGSTAB methods are required for its solution for large-scale problems. This work presents the use of GMRES solver with the MLPG method for the very first time.

In this study, the authors aim to upgrade their previous developments of the local radial basis function collocation method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic-coupled thermoelasticity problems.

The authors solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multi-quadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton–Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference finite-element method (FEM) solution and, in one case, with a solution obtained using the meshless local Petrov–Galerkin method.

The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case.

The performance of the LRBFCM near the tip of the shock front appears to be suboptimal because it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov–Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, the authors are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail.

For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity.

The purpose of this paper is to assess the performance of the stabilised moving least squares (MLS) scheme in the meshless local Petrov–Galerkin (MLPG) method for heat conduction method.

In the current work, the authors extend the stabilised MLS approach to the MLPG method for heat conduction problem. Its performance has been compared with the MLPG method based on the standard MLS and local coordinate MLS. The patch tests of MLS and modified MLS schemes have been presented along with the one- and two-dimensional examples for MLPG method of the heat conduction problem.

In the stabilised MLS, the condition number of moment matrix is independent of the nodal spacing and it is nearly constant in the global domain for all grid sizes. The shifted polynomials based MLS and stabilised MLS approaches are more robust than the standard MLS scheme in the MLPG method analysis of heat conduction problems.

The MLPG method based on the stabilised MLS scheme.

The purpose of this paper is to present a micromechanical model based on a new truly local meshless method for analysis of heat transfer in composite materials.

The presented meshless method is based on the integral form of energy equation in the sub‐particles in the material. In the presented meshless method due to elimination of domain integration the computational efforts are decreased substantially.

Numerical results are presented for temperature distribution, heat flux and thermal conductivity. Numerical results show that the presented meshless method is simple, effective, accurate and less costly method in micromechanical modeling of heat conduction in heterogeneous materials.

A small area of the composite system called representative volume element is considered as the solution domain. The fully bonded fiber‐matrix interface is considered and contact thermal resistant is neglected from the fiber matrix interface and so the continuity of temperature and reciprocity of heat flux is satisfied in the fiber‐matrix interface.

For the first time a new truly local meshless method based on the integral form of energy equation for the sub‐particles in the materials is presented for analysis of heat transfer in composite materials.

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.