MLPG Method Research Articles

Accurate and Intelligent Early Warning Method of Debris Flow Formation Based on IGWO-LSTM Algorithm

To improve the accuracy of debris flow forecasts and serve as disaster prevention and mitigation, an accurate and intelligent early warning method of debris flow initiation based on the IGWO-LSTM algorithm is proposed. First, the entropy method is employed to screen the early warning indicators. Then, the improved grey wolf algorithm (IGWO) is obtained by optimizing the grey wolf algorithm by combining elite reverse learning and adaptive convergence factors. Finally, the IGWO-LSTM algorithm is obtained by using IGWO to improve the total connection layer weight and bias parameters of LSTM, which takes the screened early warning indicators as input and outputs the early warning results of the debris flow formation risk level. In comparison with the methods introduced in earlier studies, the results demonstrate that the proposed method achieves superior outcomes in terms of assessing a single warning of multiple debris flow gullies, a multi-year warning of a single debris flow gully, and a multi-year warning of multiple debris flow gullies. The mean absolute error and root mean square error of the early warning results of the ANN model and PEEM method show low values, while the early warning hit rate shows high values, surpassing 90%. Also, the other two methods developed in the previous studies show low values of the early warning coverage rate, reaching 90% at most. Moreover, the triggered traffic model and MLPG method show high values in the early warning coverage rate, exceeding 90%, and low values in the early warning hit rate of less than 90%, and the average absolute error and root mean square error are high. On the other hand, the results of the proposed method show that the overall early warning hit rate is higher than 95%, the coverage rate is close to 100%, and the error is less than 1.5. Thus, the comprehensive analysis results show that the proposed method has better performance and higher reliability than other studied methods.

Evaluation and optimization of meshless methods to estimation of the 3D-stress intensity factors in mode I–III for fatigue life prediction cracked shaft under uni and multi-axial cyclic loading

In this research, in order to estimate 3D-Stress Intensity Factors in mode I–III, fatigue crack growth and fatigue life prediction of a cracked shaft under various cyclic loading, meshless methods are evaluated and the most appropriate method is selected. For better results, the Base Functions (BFs) are first identified and their performance and efficiency are compared with each other. In addition, while enriching the BFs in all methods and study the effect of increasing the number of sentences of Polynomial-BFs (m = 4, 7,10) and their Linearity or Quadraticity in the accuracy of calculations, two sets of Extend-Enriched-RBFs including the Multi-Quadrics and Gaussian RBFs are used in MQ-RPIM and EXP-RPIM methods. To optimize the shape parameters in the RPIM method and determine the Penalty Factor in the MLPG method, Uni and Multi-objective PSO algorithm was used. Then, a shaft with an initial semi-elliptic surface crack as a 3D-meshless domain for discretizing the weak differential equations was selected and modeled under a fixed latitude tensile, bending, and torsional cyclic loads. Changing the standard values of the shape parameters and estimating their optimal values by the PSO algorithm and comparison of findings with the results of Experimental, MLPG, PIM, FEM, and XFEM methods, has led to the best answer from the applied methods for calculation of; displacement, strain and stress fields, and the SIFs in Mode I–III. Finally, based on the results of uniaxial cyclic load analysis and selection of the MQ-RPIM method, the Multi-axial Cyclic load analysis has been performed on 3D-domain. During this analysis, the Paris Parametric equation along with the Elliptic equation and Liu’s Virtual Strain Energy (VSE) model was used to estimate the fatigue crack growth and fatigue life prediction of a submersible cracked shaft of a pump used in water pumping stations.

Geometrically Non-Linear Vibration and Coupled Thermo-Elasticity Analysis with Energy Dissipation in FG Multilayer Cylinder Reinforced by Graphene Platelets Using MLPG Method

Geometrically Non-Linear Vibration and Coupled Thermo-Elasticity Analysis with Energy Dissipation in FG Multilayer Cylinder Reinforced by Graphene Platelets Using MLPG Method

On applicability of MQ-RPIM and MLPG meshless methods with 3D extended-enriched base functions for estimation of mode I stress intensity factor and fatigue crack growth in cyclic tensile and bending load of an un-notched and notched shaft

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

A study of nonlinear systems arising in the physics of liquid crystals, using MLPG and DMLPG methods

The study of liquid crystals is one of the active areas of physics research. In this paper, the MLPG and direct MLPG (DMLPG) methods are used for the numerical study of the coupled nonlinear sine–Gordon equations in two dimensions arising in the modeling of some phenomena in liquid crystals and superconductors. To approximate numerical integrals in the local weak forms, the MLS and GMLS approximations are used in MLPG and DMLPG methods, respectively. The distribution of regular and scattered points on rectangular and irregular domains has been used to extract the numerical results. By comparing the numerical results, it can be seen that the DMLPG methods are faster, more accurate, and more efficient than the MLPG methods. These are because the GMLS approximation uses the basis polynomials instead of the complex shape functions of the MLS approximation.

DMLPG method for specifying a control function in two-dimensional parabolic inverse PDEs

This article describes a fast meshless technique, the direct meshless local Petrov–Galerkin (DMLPG) method, for identifying a control function in two-dimensional parabolic inverse PDEs. The DMLPG method is a truly meshless technique that directly approximates the local weak forms. In DMLPG, the local integrations are done over the low-degree polynomial basis functions instead of the complicated shape functions. Thus, the mass and stiff matrices are constructed by integration against polynomials. This feature overcomes the main drawback of meshless methods in comparison with the finite element methods (FEM) and significantly increases the computational efficiency in comparison with the other meshless weak form methods. In this paper, we apply the DMLPG and MLPG methods for solving two-dimensional parabolic inverse PDEs on regular and irregular domains. These methods are compared with each other and the superiority of the DMLPG over the classical MLPG is demonstrated. The numerical results confirm the good efficiency of the DMLPG method for solving inverse problems especially coefficient inverse problems. Finally, we will conclude that the DMLPG method can be considered as an attractive alternative to existing meshless weak form methods in solving inverse problems.

Numerical approximation of the fractional HIV model using the meshless local Petrov\u2013Galerkin method

This paper deals with the model of fractional HIV-1 infection of CD4+T cells transformation with homogeneous Neumann boundary conditions. Numerical methods for solving fractional time differential equations are developed with Caputo’s definition. The forward difference methods were constructed applied to the approximation of the fractional time differential equation. The MLPG method is used to solve the problem of fractional HIV models for spatial discretization. Approximated solutions at the time level n use conventional iterative methods such as fixed point iterations to handle the nonlinear parts. An analysis of stability and convergence of numerical schemes is presented along with the eigenvalue of the matrix. The abilities of the developed formula was confirmed through four numerical examples base on convergence and accuracy of numerical results. The results of the numerical experiments were compared with the solution of the integer order differential equation to confirm the accuracy and efficiency of the proposed scheme. The simulation results show that the formula is easy to use and useful for those interested in fractional derivatives.

Predicting fatigue crack propagation in residual stress field due to welding by meshless local Petrov-Galerkin method

This paper introduces a novel procedure to develop the MLPG method for investigating residual stress effect on the Fatigue Crack Propagation (FCP) rate. A new formulation is introduced based on thermoelastic-plastic equation for this method to numerical analysis of the residual stress due to welding. The most important part of numerical analysis by MLPG method is to determine the residual stress redistribution due to crack growth and calculation of the Stress Intensity Factor (SIF) in residual stress field. A good agreement is seen between the outputs of MLPG method to the Hole-Drilling Strain-Gage method results. The standard weight function is developed without increasing the computational time for simulation of the displacement and stress around the crack. The Superposition principle is employed to consider the residual stress effect on the SIF and cycle ratio. Finally, the Walker’s FCP equation is modified to take into account the simultaneous effects of cyclic loading and residual stress. It was discovered that the results obtained from the purposed method is in a good agreement with FCP experimental results. Therefore, it can be concluded a new approach is developed to analyze the calculation of SIF in the residual stress field and its effect on the FCP rate.

Quasi-static analysis of mixed-mode crack propagation using the meshless local Petrov–Galerkin method

The efficiency of classical truly meshless local Petrov–Galerkin method with linear test function approximation in the crack growth problems of complex configurations is investigated in this article. Several unique and multiple (two) cracks are examined to show the accuracy of the proposed meshfree method by comparing the crack path with boundary element, finite element, element-free Galerkin and experimental results. Crack propagation under both mechanical and thermal loads and also mode I and mixed-mode conditions are evaluated. Effect of the functionality of material properties on crack paths and sensitivity of crack growth to boundary conditions are investigated. Classical maximum circumferential stress criterion is employed into the formulation of the MLPG method to predict crack growth direction. Despite the common numerical studies, a simple and straightforward automatic increment size determination method is introduced and implemented in the MLPG method. Besides, regular domain point distribution in the vicinity of the crack tip without the increased density of points is used. Various test problems show the computational accuracy and applicability of this method to linear elastic fracture mechanics problems.

Two-level meshless local Petrov Galerkin method for multi-dimensional nonlinear convection–diffusion equation based on radial basis function

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method. Two-level method is employed to save the computation time, solving one small linearized convection–diffusion equation in a coarse nodal distribution by Newton iterative scheme and then processing one linear equation on a fine nodal distribution. The convergence analysis for the MLPG method and the two-level method is proven by applying the RBF interpolation method. Finally, the numerical examples provide a sufficient support and show that the proposed method is efficient for solving nonlinear convection–diffusion equation.

Bending analysis of FG plates using a general third-order plate theory with modified couple stress effect and MLPG method

Meshless local Petrov–Galerkin analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect is presented. Governing equations of problem are a fourth-order partial differential equations system which derived in terms of eleven generalized displacement variable, by applying the principle of virtual displacements. The moving least-squares approach is used for approximation of unknown variables and the Gauss weight function is employed as test function for obtaining local weak form. The Gauss–Legendre quadrature method is utilized for numerical integration of weak equations. Static bending results of a simply-supported plate is obtained for various power law index and length scale parameter, and is compared to analytical solutions that shows high accuracy in results.

Direct meshless local Petrov–Galerkin (DMLPG) method for 2D complex Ginzburg–Landau equation

In this paper, the direct meshless local Petrov–Galerkin (DMLPG) approximation is proposed to solve the two-dimensional complex Ginzburg–Landau equation. DMLPG uses a generalized MLS (GMLS) method which directly approximates test functionals by values at nodes and therefore avoids integration over MLS shape functions and replaces it by a much cheaper integration over polynomials. An Implicit-Explicit method used to deal with the time derivative. Several examples are performed to compare the accuracy and efficiency of the DMLPG method with the traditional MLPG method.

Transient Dynamic Stress Intensity Factor of FGM Plates Using the State Space and MLPG Methods

In this paper, a new effective method named state-space is extended and utilized to analyze FGM dynamic fracture problems, discretized by the meshless local Petrov–Galerkin method. Both the moving least square and the direct method have been applied to estimate the shape function and to impose the essential boundary conditions. The visibility criterion is used to simulate the displacement and stress field around the crack tip. Normalized dynamic stress intensity factors are calculated using the path-independent integral, J*, which is formulated for the non-homogeneous material. Two methods commonly used in solving such discretized problems, namely, the central difference method and the Newmark method, are compared with the proposed method. Both homogeneous and non-homogeneous (FGM) center-cracked plates under uniform tensile impact load are used to examine the accuracy and the computational time of the new method. The results of the implementation of these methods are compared with existent solutions. The results show that the state-space method saves substantial computational time for a given accuracy. The center-cracked plate made of FGM with two different material gradations (along and normal to the crack length) and four different FGM zones under the effect of uniform tensile impact load are considered, and the dynamic stress intensity factors are studied under the influence of various non-homogeneity ratios.

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in 3D complex geometry

Development of meshfree methods is in full swing in the last few decades as these methods overcome the shortcomings of mesh based methods. Development of meshfree methods for large scale problems requires fast iterative solvers. Preconditioner plays an important role in improving convergence rate of an iterative solver. Recently proposed SRJ method can be used as a stand-alone solver and as a preconditioner in CG type methods. Objective of the present work is to evaluate the performance of SRJ method as a preconditioner in BiCGSTAB method in connection with the MLPG method. Three dimensional Poisson equation and heat conduction equation are solved in regular and irregular domain respectively. The SRJ method is tested as a solver and as a preconditioner in BiCGSTAB method. Its performance as a preconditioner is compared with Jacobi and SOR preconditioners.

Quick and robust meshless analysis of cracked body with coupled generalized hyperbolic thermo-elasticity formulation

In this article, the MLPG method is applied to the generalized linear coupled thermoelectricity equations. Lord–Shulman modification with a relaxation time parameter is used in the hyperbolic heat conduction equations. A new linear test function which is zero on the boundaries of local test domains is introduced. The test function and its partial derivatives are determined by an exponential RBF approximation method. The approximation of test function and main variables are similar. For the construction of shape functions, neighbors of every point are determined based on the definition of the closest adjacent point pattern. Consequently, test function space becomes independent of trial function space. Direct interpolation method and penalty parameter are used to impose essential boundary conditions. The selection of appropriate parameters are demonstrated in two numerical examples. The small number of used points is the advantage of this method over the FEM that is shown in several examples. The accuracy of results is compared between the meshless method and different analytical and FE solutions. The effect of the relaxation time on SIF under thermal shock is discussed in a separate example. The comparison of meshless results with various examples shows that employed method is accurate and reliable.

A greedy MLPG method for identifying a control parameter in 2D parabolic PDEs

In this paper, we consider coefficient inverse problems, which are associated with the identification of unknown time dependent control parameter and unknown solution of two-dimensional parabolic inverse problem with overspecialization at a point in the spatial domain. After suitable finite difference approximation of time variable, an MLPG method is used for spatial discretization. To improve the efficiency of the MLPG method, a greedy algorithm is used. In fact, using the greedy algorithm, we avoid using more points from the data site than absolutely necessary and therefore, the method becomes more efficient. Comparison of the different kind of point selection and the effect of noisy data are performed for four test problems while our last test problem considers a problem with unknown solution. The results reveal that the method is efficient.

Dynamic analysis of non-symmetric FG cylindrical shell under shock loading by using MLPG method

The Dynamic equations in the polar coordinates are drawn out using the MLPG method for the non-symmetric FG cylindrical shell. To simulate the mechanical properties of FGM, the nonlinear volume fractions for radial direction are used. The shape function applied in this paper is a form of the radial basis functions, by using this function all the requirements for an effective and suitable shape function are established. Hence in this study, the multiquadrics (MQ) radial basis functions are exploited as the shape function governing the problem. The MLPG method is combined with the the Newmark time approximation scheme to solve dynamic equations in the time domain. The obtained results by the MLPG method to be verified are compared with the analytical solution and the FEM. The obtained results through the MLPG method show a good agreement in comparison to other results and the MLPG method has high accuracy for dynamic analysis of the non-symmetric FG cylindrical shell. To demonstrate the capability of the present method to dynamic analysis of the non-symmetric FG cylindrical shell, it is analyzed dynamically with different volume fraction exponents under harmonic and rectangular shock loading. The present method shows high accuracy, efficiency and capability to dynamic analysis of the non-symmetric FG cylindrical shell with nonlinear grading patterns.

Nonlinear numerical analysis of convective-radiative fin using MLPG method

Nonlinear numerical analysis of convective-radiative fin using MLPG method

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in complex geometry

ABSTRACTMeshless local Petrov–Galerkin (MLPG) method is a promising meshfree method for continuum problems in complex domains, especially for large deformation, moving boundary and phase change problems. For large-scale problems, iterative methods for solving the discretized equations are more suitable than direct methods. Krylov subspace solvers of conjugate gradient type are the most preferred iterative solvers. The convergence rate of these methods depends on preconditioner used. Recently, proposed schedule relaxation Jacobi (SRJ) method can be used as a stand-alone solver and as a preconditioner. In the present work, the SRJ method is tested as a stand-alone solver and as a preconditioner for BiCGSTAB solver in the MLPG method, and its performance has been compared with successive overrelaxation (k) preconditioner. Two-dimensional linear steady-state heat conduction in complex shape geometry has been used as the model test problem.

A local meshless method based on the finite collocation and local integral equations method for delay PDEs

A local meshless method based on the finite collocation, local radial basis function (RBF) and MLPG method is proposed for solving the 2D Delay partial differential equations. In fact, instead of collocation of governing equations in the finite collocation method, we propose the use of local weak form of governing equations on the local stencils. The method is used for the numerical solutions of the 2D delay partial differential equations (PDEs). Two delay and one nonlinear multidelay parabolic PDEs are solved as test problems and the comparisons of the results reveal the effectiveness of the method.