Meshless Local Petrov-Galerkin Research Articles

INverse problem of identifying a time-dependent coefficient and free boundary in heat conduction equation by using the meshless local Petrov-Galerkin (MLPG) method via moving least squares approximation

In this paper we investigated the inverse problem of identifying an unknown time-dependent coefficient and free boundary in heat conduction equation. By using the change of variable we reduced the free boundary problem into a fixed boundary problem. In direct solver problem we employed the meshless local Petrov-Galerkin (MLPG) method based on the moving least squares (MLS) approximation. Inverse reduced problem with fixed boundary is nonlinear and we formulated it as a nonlinear least-squares minimization of a scalar objective function. Minimization is performed by using of f mincon routine from MATLoptimization toolbox accomplished with the Interior - point algorithm. In order to deal with the time derivatives, a two-step time discretization method is used. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model

The present paper outlined a procedure for geometrically nonlinear dynamic analysis of functionally graded graphene platelets-reinforced (GPLR-FG) nanocomposite cylinder subjected to mechanical shock loading. The governing equation of motion for large deformation problems is derived using meshless local Petrov-Galerkin (MLPG) method based on total lagrangian approach. In the MLPG method, the radial point interpolation technique is employed to construct the shape functions. A micromechanical model based on the Halpin-Tsai model and rule of mixture is used for formulation the nonlinear functionally graded distribution of GPLs in polymer matrix of composites. Energy dissipation in analyses of the structure responding to dynamic loads is considered using the Rayleigh damping. The Newmark-Newton/Raphson method which is an incremental-iterative approach is implemented to solve the nonlinear dynamic equations. The results of the proposed method for homogenous material are compared with the finite element ones. A very good agreement is achieved between the MLPG and FEM with very fine meshing. In addition, the results have demonstrated that the MLPG method is more effective method compared with the FEM for very large deformation problems due to avoiding mesh distortion issues. Finally, the effect of GPLs distribution on strength, stiffness and dynamic characteristics of the cylinder are discussed in details. The obtained results show that the distribution of GPLs changed the mechanical properties, so a classification of different types and volume fraction exponent is established. Indeed by comparing the obtained results, the best compromise of nanocomposite cylinder is determined in terms of mechanical and dynamic properties for different load patterns. All these applications have shown that the present MLPG method is very effective for geometrically nonlinear analyses of GPLR-FG nanocomposite cylinder because of vanishing mesh distortion issue in large deformation problems. In addition, since in proposed method the distributed nodes are used for discretization the problem domain (rather than the meshing), modeling the functionally graded media yields to more accurate results.

Meshless method – Review on recent developments

The present article outlines the historical development of Meshless method. The high computational cost associated with conventional numerical methods and their lack of accuracy, particularly in the area of astrophysics is the motivation and an initiation towards the invention of latest meshless methods. The review starts with Smoothed particle hydrodynamics meshless method which was invented by Gingold, Monghan in 1977 for astrophysics applications. Furthermore, shortcomings of the SPH method, it’s solution, and recent development has been reviewed for different applications. After that advent features of Element free Galerkin method with mean least square method and Lagrangian multiplier have been critically reviewed for different cases. Later, the radial basis function method is explained along with its extended version to solve a partial differential equation. Many types of RBF methods have been compared. Also, several consistency and accuracy problems are reviewed for Reproducing Kernel Particle Method. Recent advancement like generalization of RKPM method, hybrid of RKPM with Lagrangian multiplier has been briefed for a different approach. Additionally, feasibility, flexibility and several applications of Meshless local Petrov Galerkin have been reviewed. Lastly recent development of MLPG with Laplace transform to solve PDE have been studied. Afterwards, a different attribute of point collocation method for inherent multiresolution capability and error estimation has been reviewed for different applications. Briefly, the article provides a condensed overview of several meshless methods, its recent advancement and their applications.

Modifying a meshless method to solving κ−ε turbulent natural convection heat transfer

The conventional meshless local Petrov–Galerkin method is modified to enable the method to solve turbulent convection heat transfer problems. The modifications include developing a new computer code which empowers the method to adopt nonlinear equations. A source term expressed in terms of turbulent viscosity gradients is appended to the code to optimize the accuracy for turbulent flow domains. The standard [Formula: see text] transport equations, one of the most applicable two equation turbulent viscosity models, is incorporated, appropriately, into the developed code to bring about both versibility and stability for turbulent natural heat transfer applications. The amenability of the new developed technique is tested by applying the modified method to two conventional turbulent fluid flow test cases. Upon the obtained acceptable results, the modified technique is, next, applied to two conventional natural heat transfer test cases for their turbulent domain. Based on comparing the results of the new technique with those of the available experimental or conventional numerical methods, the proposed method shows good adaptability and accuracy for both the fluid flow and convection heat transfer applications in turbulent domains. The new technique, now, furthers the applicability of the mesh-free local Petrov-Galerkin (MLPG) method to turbulent flow and heat transfer problems and provides much closer results to those of the available experimental or conventional numerical methods.

Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains

A new numerical scheme has been developed based on the fast and efficient meshless local weak form i.e direct meshless local Petrov–Galerkin (DMLPG) method for solving the fractional fourth-order partial differential equation on computational domains with complex shape. The fractional derivative is the Riemann–Liouville fractional derivative. At first, a finite difference scheme with the second-order accuracy has been employed to discrete the time variable. Then, the DMLPG technique is employed to achieve a full-discrete scheme. The time-discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the L2 space. Also, some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non-regular grid points.

Computational Analysis of Linear Elastic Crack Growth in Functionally Graded Bodies Using Non-Uniform Steps Integrated in the MLPG

This paper describes the estimation of mixed-mode crack growth path in functionally graded materials under mechanical and thermal loads by the meshless method. Simple and complex geometries with single and double cracks are provided to verify the merits of a recently proposed modified meshless local Petrov–Galerkin method and to show the applicability of this method in crack growth simulation. Polygon test function domains are constructed locally at every crack growth step in order to numerically calculate contour and domain integrals. The classical form of thin plate spline radial basis function without enrichment with a regular distribution of points in the vicinity of crack tips are used. In this efficient method, the total number of domain points are kept fixed during crack growth steps forever. For this end, a simple method of new crack tip determination at every growth step is proposed. The MLPG results are compared with reference solutions including experimental, finite element, boundary element and other meshless methods and the accuracy of this method is investigated in different numerical test problems.

A study of meshless methods for optimization of cathodic protection systems

Over the last decades, several computer codes have been developed aiming at the full 3D simulation of cathodic protection (CP) systems. CP is a technique applied to prevent corrosive processes and the main goal of the simulation has been to predict the degree of corrosion control achieved. Many pioneering works allowed for the successful application of the boundary element method (BEM) to CP systems. The aim of the present contribution is to introduce a brief overview of cathodic protection system modelling, including some numerical simulations. Mathematical formulations for the electrochemical potential problem are proposed, considering the following meshless methods: the method of fundamental solutions (MFS) and a meshless local Petrov-Galerkin (MLPG2) procedure. The meshless methods performances are evaluated comparing their results with a direct BEM solution procedure. The meshless applications are original first time attempts of such formulations to corrosion problems and cover a lot of practical situations found in actual cathodic protection applications.

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.

The application of the meshless local Petrov-Galerkin method for the analysis of heat conduction and residual stress due to welding

The structural analysis of welding based on the thermal-elastic-plastic finite element method can lead to acceptable accuracy for the short time of welding. But if the welding time is long and the thermal source is moving, problems arise in the analysis due to the mesh re-generation. In this paper, in order to overcome the weak points of the finite element method, the meshless local Petrov-Galerkin (MLPG) method by using moving least square (MLS) has been used, based on the weak solution of equations governing the temperature field and the residual stress caused by welding. Without increasing the computational time, accuracy of the MLPG method has increased by developing this numerical method by taking into account the influence of MLPG parameters. To investigate the distance and density of nodal distribution, the relation between support domain size and quadrature domain size with distance between nodes is optimized. A good agreement is seen between the outputs of this numerical method to the hole-drilling strain-gauge method results. Therefore, it can be concluded that a new application based on thermoelastic-plastic equation for the MLPG method to numerical analysis of the residual stress due to welding is introduced.

Nonlinear analysis of convective-radiative longitudinal fin of various profiles

Purpose This paper aims to study nonlinear heat transfer through a longitudinal fin of three different profiles. Design/methodology/approach A truly meshfree method is used to undertake a nonlinear analysis to predict temperature distribution and heat-transfer rate. Findings A longitudinal fin of three different profiles, such as rectangular, triangular and concave parabolic, are analyzed. Temperature variation, along with the fin length and rate of heat transfer in steady state, under convective and convective-radiative environments has been demonstrated and explained. Moving least square (MLS) approximants are used to approximate the unknown function of temperature T(x) with Th(x). Essential boundary conditions are imposed using the penalty method. An iterative predictor–corrector scheme is used to handle nonlinearity. Research limitations/implications Modelling fin in a convective-radiative environment removes the assumption of no radiation condition. It also allows to vary convective heat-transfer coefficient and predict the closer values to the real problems for the corresponding fin surfaces. Originality/value The meshless local Petrov–Galerkin method can solve nonlinear fin problems and predict an accurate solution.

Prediction of Groundwater Fluctuations Using Meshless Local Petrov-Galerkin Numerical Method in a Field Aquifer (Birjand Aquifer)

Prediction of Groundwater Fluctuations Using Meshless Local Petrov-Galerkin Numerical Method in a Field Aquifer (Birjand Aquifer)

Application of meshless local Petrov–Galerkin technique to simulate two-dimensional time-fractional Tricomi-type problem

The paper is devoted to investigate the two-dimensional time-fractional Tricomi-type equation, which describes the anomalous process of nearly sonic speed gas dynamics. An efficient numerical process, based on the combination of time stepping method and meshless local weak formulation, is performed to solve the model. Firstly, an implicit finite difference scheme is used to discrete the problem in time direction. The unconditional stability of the proposed time discretization scheme is proven. Then, a meshfree method based on the combination of local Petrov–Galerkin formulation and strong form is implemented to fully discretize the underlying problem. In our implementation, the radial point interpolation basis functions and local Heaviside step functions are used as the basis and test functions, respectively. A simple collocation process is employed to impose the Dirichlet boundary conditions directly. Finally, two numerical experiments on regular and irregular domains are presented to verify the efficiency, validity and accuracy of the technique.

Active tuning and maximization of natural frequency in three-dimensional functionally graded shape memory alloy composite structures using meshless local Petrov–Galerkin method

A mesh-less method based on the local Petrov–Galerkin approach is developed and numerically implemented for active tuning and three-dimensional (3D) analysis of free vibration response of functionally graded (FG) shape memory alloy (SMA) sandwich composites. The local symmetric weak formulation is derived based on the 3D equilibrium equations of elasticity and the field variables are interpolated using the moving-least squares approach. Two main goals are followed. Firstly, free vibration analysis of a sandwich plate made of a SMA fiber reinforced composite core and two FG face sheets are presented. A parametric study is performed to investigate the influences of design parameters on the fundamental natural frequency (FNF) of composite structure and also maximize that. Secondly, SMA wires are embedded to fiber reinforced (FR) composite structures to enable precisely active control of their dynamic characteristics and change that in a required manner. To increase the controllability level and tune the complex exhibits, functionally graded shape memory alloys are considered which have the advantage of combining the functionalities of the shape memory effect and those of FG structures. This paper uses multi-layer activation patterns and the polynomial graded orientation angle of SMA wire along the structure thickness to furnish the FR composite designer with a powerful free vibration control tool. Finally, two activation techniques of SMA wires, the active property tuning (APT) and the active strain energy tuning (ASET) techniques, are compared. It is concluded from all investigated cases that the obtained FNF of ASET technique is always larger than the APT technique.

Stabilised MLS in MLPG method for heat conduction problem

PurposeThe 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.Design/methodology/approachIn 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.FindingsIn 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.Originality/valueThe MLPG method based on the stabilised MLS scheme.

An efficient meshless computational technique to simulate nonlinear anomalous reaction–diffusion process in two-dimensional space

In this paper, a numerical simulation of an anomalous reaction–diffusion process in two-dimensional space with a nonlinear source term is presented. An efficient and powerful computational technique based on a combination of a time integration scheme and meshless local Petrov–Galerkin method is implemented for solving the problem. An implicit time stepping scheme with second-order accuracy is used to discretize the problem in the time direction. The unconditional stability of the proposed time discretization scheme is proved in an appropriate Sobolev space. In order to obtain a fully discrete model, the primary spatial domain is represented by a set of regularly distributed nodes and spatial shape functions are constructed on the distributed field nodes using the radial point interpolation method. Then, a local weak form meshless method based on the radial point interpolation shape functions is used to discretize the problem in the spatial direction. The efficiency and accuracy of the proposed computational procedure are demonstrated by numerical examples. Numerical results show that the method approximates solutions of the model in excellent agreement with analytical results. Moreover, it can be observed that the method gives the expected second-order temporal convergence rate independent of the fractional order which confirms the theoretical predictions.

A New Test Function for Acoustic Computations with Meshless Methods

The meshless local Petrov–Galerkin (MLPG) and the local boundary integral equation (LBIE) methods has been introduced approximately three decades ago. These methods are based on writing the local weak form of the governing equation and performing subsequent numerical integration and interpolations in the local subdomains. A key step is the choice of the test function in the local weak form, which has historically led to several different formulations regarding the final form of the local integrals. Considering the application of the methods to acoustics, four different test functions have been employed so far in the literature; all of these approaches resulted in formulations which contain domain integrals. In this paper, we present a new test function to be used in meshless methods, which yields a simple form of the local integral equation without domain integrals and provides significant improvement in terms of the computational time and CPU requirements. The efficiency and the accuracy of the new method are presented and compared with the previous methods.

Numerical simulation of mixed convection in a square cavity using Direct Meshless Local Petrov-Galerkin (DMLPG) method.

This paper present the extension of the Direct Meshless Petrov-Galerkin (DMLPG) for solving mixed convection heat transfer in a square cavity for the flow up to Reynolds number up to Re 400 and Grashof number Gr = 100. The semi algebraic forms of the governing equations are solved using fractional step method. The numerical results of the DMLPG method are in good agreement with the conventional numerical results in the literature. Based on the numerical results it is shown that The DMLPG is very promising in dealing with the fluid flow and heat transfer problem.

Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries

We propose a three dimensional meshless numerical scheme based on the Boltzmann equation for the simulation of fluid flows in domains of complex geometries. The velocity space discretization in our method is the same as the standard lattice Boltzmann method, i.e. the same lattices are used to discretize microscopic velocities. The velocity-discrete Boltzmann equation with the BGK collision approximation is discretized in time using the Lax–Wendroff scheme, and subsequently in space using the meshless local Petrov–Galerkin method based on local radial basis functions augmented with polynomials. The method is first validated and tuned by solving two benchmark problems: The pressure-driven flow in an elliptical tube, and the three-dimensional lid-driven cavity flow. The error analysis for the first case shows the second order accuracy of the proposed method. The flow in a packed bed is then solved in order to illustrate the efficiency of the proposed method in the simulation of geometrically complex flows. The results show that both the computational time and the memory usage of our method are lower than those of the standard lattice Boltzmann method for the geometrically complex flows.