Direct Meshless Local Petrov–Galerkin Research Articles

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

A truly meshless approach to structural topology optimization based on the Direct Meshless Local Petrov–Galerkin (DMLPG) method

A truly meshless approach to structural topology optimization based on the Direct Meshless Local Petrov–Galerkin (DMLPG) method

Fourth-order exponential time differencing Runge–Kutta scheme and local meshless method to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials

The current article is devoted to proposing a local meshfree method for simulating the unsteady diffusion–convection problems of anisotropic functionally graded materials (AFGM). The first stage of the proposed numerical algorithm is based on discretizing the spatial variables by utilizing the direct meshless local Petrov–Galerkin (DMLPG) method. Then a system of ODEs is extracted which is associated to the time variable. Furthermore, the fourth-order exponential time differencing Runge–Kutta method is applied to solve the system of ODEs. The main aim of this work is to introduce a flexible and low-cost numerical procedure to solve unsteady diffusion–convection problems of AFGM on the complicated geometry. The new numerical technique has admissible resultants to simulate the studied model. This numerical formulation is applied to several examples with complex domains to check its ability.

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

Simulation of predator–prey system with two-species, two chemicals and an additional chemotactic influence via direct meshless local Petrov–Galerkin method

PurposeThis 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.Design/methodology/approachIn 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.FindingsThis current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.Originality/valueThe proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

The direct meshless local Petrov–Galerkin technique with its error estimate for distributed-order time fractional Cable equation

Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov–Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the M-point Gauss–Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(τ2−max{α,β}) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.

Numerical study of the variable-order time-fractional mobile/immobile advection-diffusion equation using direct meshless local Petrov-Galerkin methods

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O(Δt2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

In this paper, we employ an improved Meshless Local Petrov–Galerkin (MLPG), namely Direct Meshless Local Petrov–Galerkin (DMLPG) method, for solving an inverse time-dependent source problem for two-dimensional fractal mobile/immobile solute transport equation on regular and irregular domains. In the weak form DMLPG method, numerical integrations apply over low-degree polynomial basis functions instead of complicated moving least squares (MLS) shape functions of MLPG method. Therefore, the computational costs often reduce in the DMLPG method. In space domain, we employ the DMLPG method and the time derivatives of problem are discretized based on finite difference schemes. Numerical results demonstrate efficiency and accuracy of the proposed algorithm for solving the inverse source problem.

Numerical study of the unsteady 2D coupled magneto-hydrodynamic equations on regular/irregular pipe using direct meshless local Petrov–Galerkin method

The study of the conduction of fluids in magnetic fields has attracted a lot of interest in recent years. In this paper, we use new truly meshless methods for numerical solving of the unsteady 2D coupled magneto-hydrodynamic (MHD) equations with uniform and scattered distributions of nodes on regular and polar domains. These methods are implemented based on a generalization of the moving least squares approximation (GMLS) method and local weak forms of MHD equations. By observing the numerical results obtained from these methods and comparing them with others, one can find the efficiency, accuracy, and speed of these methods.

A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrödinger system appearing in Bose–Einstein condensates

In this work, one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) coupled damped Schrödinger system is solved numerically. A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed. Polyharmonic splines are used as radial basis function with augmented polynomials. The use of the polyharmonic splines saves us from choosing an optimum shape parameter which is not a simple task for infinitely smooth RBFs such as multiquadrics or Gaussians. For time discretization classical fourth-order Runge Kutta method is utilized. L∞ error norm and conserved quantities are computed to indicate performance of the proposed method. Stability of the proposed method is examined numerically. Some computer codes are devised in Julia programming language for obtaining numerical results. Acquired numerical results and their comparison with other studies available in literature such as cubic B-spline Galerkin method and direct meshless local Petrov–Galerkin (DMLPG) method endorse the performance and reliability of the proposed method.

Numerical solution strategy for natural convection problems in a triangular cavity using a direct meshless local Petrov-Galerkin method combined with an implicit artificial-compressibility model

Thermal analysis of natural convection in a cavity is solved by numerical approach. The direct meshless local Petrov–Galerkin method combined with an implicit artificial-compressibility model is proposed here to simulate laminar natural convective heat transfer in triangular cavities. The spatial terms of the governing equation are discretized using the direct meshless local Petrov–Galerkin method with the Dirac function as the test function. The method uses an artificial-compressibility scheme to couple the pressure with the velocity components directly. The semi-algebraic system is solved using an implicit backward Euler pseudo-time method. Kovasznay flow problem is presented to examine the accuracy of proposed method. The validation of the obtained results by using experimental data is also provided, and four test problems — natural convection in an equilateral triangular cavity, a right-angled triangular cavity, an isosceles triangular cavity, and an irregular triangular cavity — are solved by using the proposed method. The numerical results obtained using the proposed method showed excellent agreement with those obtained using the conventional methods available in the literature.

Investigation of generalized Couette hydromagnetic flow of two-step exothermic chemical reaction in a channel via the direct meshless local Petrov–Galerkin method

This paper concerns to propose a high-order local meshless method to simulate the Couette flow based on the incompressible Navier–Stokes equation and the generalized Couette hydromagnetic flow of two-step exothermic chemical reaction in a channel. The numerical solution is performed in the two main stages. First, time variable of the incompressible Navier–Stokes equation is approximated by a finite difference scheme as in this procedure, the velocities in x- and y-directions must be updated by using an explicit difference formula. Also, the pressure component is calculated by solving a Poisson’s equation in each time step. Here is the second stage e.g. a local meshless method based on the direct meshless local Petrov–Galerkin method is employed to solve the Poisson’s equation which corresponds to the pressure value. Some famous phenomena such as Couette flow are studied by using the proposed numerical algorithm. The results show that the new numerical procedure not only has suitable and acceptable accuracy but also it simply solves such interesting models. Up to the best of our knowledge, this is the first time that the direct meshless local Petrov–Galerkin method is applied for these problems and it can have more accuracy than other meshless methods.

An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems

In this work, we propose an efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale approach for stabilizing numerical solutions. It is a well known fact that using polynomial basis without any modifications to obtain numerical solutions of partial differential equations may be peculiar owing to ill conditioned resultant coefficient matrix which is formed after process of discretization. Hence, as a remedy to the highly ill-conditioned coefficient matrix we employ multiple-scale approach. The main idea behind the multiple-scale approach is to make norm of all columns of resultant coefficient matrix equal to each other. The proposed method is a truly strong-form meshfree method since we do not need any mesh or integration process in problem domain, these features makes the implementation of the method very simple in computer environment. The efficiency of the proposed method is tested by some test problems which may have smooth interface or interface with sharp corners. Stability of the proposed method is investigated numerically in the presence of noise effect. Further, to show accuracy of the proposed method we present some comparisons with available numerical methods in literature, such as direct meshless local Petrov-Galerkin method, matched interface and boundary method, spectral element method and some meshless methods based on radial basis functions. The obtained numerical results and their comparisons confirm applicability of the proposed method for 2-D and 3-D steady state elliptic interface problems.

Direct meshless local Petrov-Galerkin method to investigate anisotropic potential and plane elastostatic equations of anisotropic functionally graded materials problems

The meshless local technique is employed to simulate potential problems involving material anisotropy and also plane elastostatic equations of anisotropic functionally graded materials (PEE-AFGM). We utilize direct meshless local Petrov Galerkin (DMLPG) technique to discrete the spatial directions. The DMLPG5 (based on weak form) is applied to solve potential problems involving material anisotropy and the DMLPG2 (based on strong form) is employed to simulate PEE-AFGM. The coefficients of the anisotropic conductivity require the gradient term to be modified accordingly. The grading function can be any general function that varies smoothly from point to point in the material. The developed procedures are tested on various examples on non-rectangular domains to check their accuracy.

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.

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.

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.

A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations

In this study, the direct meshless local Petrov–Galerkin (DMLPG) method has been employed to solve the stochastic Cahn–Hilliard–Cook and Swift–Hohenberg equations. First of all, we discretize the temporal direction by a finite difference scheme. In order to obtain a fully discrete scheme the direct meshless local collocation method is used to discretize the spatial variable and the Euler–Maruyama method is used for time discretization. The used method is a truly meshless technique. In order to illustrate the efficiency and accuracy of the explained numerical technique, we study two stochastic models with their applications in biology and engineering, i.e., the stochastic Cahn–Hilliard–Cook equation and a stochastic Swift–Hohenberg model.

DMLPG method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear Schrödinger system which arises from Bose–Einstein condensates

In this paper, the direct meshless local Petrov–Galerkin (DMLPG) method is applied to find the numerical solution of coupled damped nonlinear Schrödinger system in one, two and three-dimensional spaces. The propagation properties of single soliton, double and triple solitons of coupled damped nonlinear Schrödinger system are simulated and the interactions between these solitons are studied numerically. The efficient time differencing Runge–Kutta method is utilized for the time discretization. DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions and this significantly increases the computational efficiency of DMLPG in comparison with the other meshless local weak form methods especially in two and three dimensions. The main aim of this paper is to show that the DMLPG method can be simply used for solving high-dimensional system of non-linear partial differential equations especially coupled damped nonlinear Schrödinger system. The numerical results confirm the good efficiency of the proposed method for solving our model.

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.