Meshless Petrov-Galerkin Method Research Articles

Effectiveness of meshless methods for advection dominant groundwater contaminant transport problems

Meshless methods are efficient tools in prediction of fate of groundwater contamination. In these methods, the choice of amount of discretization in space and time affects the accuracy of solution and computational time. Peclet number is a controlling parameter of such numerical transport simulation models and is the measure of space discretization adequacy for a given velocity and dispersion coefficient. Greater value of Peclet number indicates highly advective flow with negligible diffusion. In this study, the effect of Peclet number on accuracy of meshless methods is analysed. Three meshless methods of three different categories of formulation are considered, namely, Radial Point Collocation Method (RPCM), a meshless strong form method, weak form Meshless Petrov Galerkin Method (MLPG) and hybrid RPCM-MLPG Meshless Weak Strong (MWS) form method. These are applied for modelling Groundwater Contaminant Transport (GCT) phenomenon. The three models are demonstrated using hypothetical case studies and the results are verified with analytical solutions. After a successful validation, the models are subjected to changes in Peclet number and the errors with respect to analytical solutions are studied. It is found that MLPG-GCT model is the most stable and MWS-GCT model is more accurate within less computational time for reasonably high Peclet numbers.

Numerical solution of the Richards equation in unsaturated soil using the meshless Petrov–Galerkin method

Nowadays, infiltration tactics are widely used to manage storm water in urban areas. These techniques are used and recognized around the world due to their benefits, such as reducing the negative consequences of urbanization, reducing storm water flow in sewage systems and recharging groundwater. Richards equation is one of the most well-known equations for simulating water infiltration in soil in the unsaturated area. In the present study, a one-dimensional approach to the numerical solution of Richards equation is presented using meshless Petrov–Galerkin method and Kirchhoff transformation. The results of this modeling have been compared with analytical solution, laboratory data, and finite difference and meshless numerical methods. Given that the proposed model can provide an accurate representation of water level changes in unsaturated soil compared to analytical solution, laboratory data, finite difference method and MQ-RBF with the root mean square error equal to 0.09, 1.02, 0.7 and 0.1, it can be claimed that the model can model the flow of water infiltration in unsaturated soil.

A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary condition

This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov–Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov–Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.

A new stability parameter in streamline upwind meshless Petrov–Galerkin method for convection–diffusion problems at large Peclet number

The numerical oscillation will occur in convection–diffusion equations as special linear problems at large Peclet number (Pe) in the numerical calculation process. In this article, we propose a new definition of the stability parameter in streamline upwind meshless Petrov–Galerkin (SUMLPG) method. The most important feature of the proposed method is that the test function in the stabilization term is taken into the differential operator-like form The stability parameter is designed to adjust the convection strength to achieve accurate and stable numerical solutions. Several classical examples are adopted to assessment the accuracy and stability of the proposed stability parameter. It is proven that the proposed method is especially suitable for convection–diffusion problems with large Pe.

Bending analyses of 1D orthorhombic quasicrystal plates

The meshless Petrov–Galerkin method (MLPG) is applied to plate bending analysis in 1D orthorhombic quasicrystals (QCs) under static and transient dynamic loads. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation plate theory depends only on the in-plane coordinates on the mean plate surface. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. The spatial variations of the phonon and phason displacements are approximated by the moving least-squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations (ODEs) for nodal unknowns is obtained. The system of the ODEs of the second order is solved by the Houbolt finite-difference scheme. Our numerical examples demonstrate clearly the effect of the coupling parameter on both static and dynamic phonon/phason deflections.

Taylor's Meshless Petrov-Galerkin Method for the Numerical Solution of Burger's Equation by Radial Basis Functions

During the last two decades, there has been a considerable interest in developing efficient radial basis functions (RBFs) algorithms for solving partial differential equations (PDEs). In this paper, we introduce the Petrov-Galerkin method for the numerical solution of the one-dimensional nonlinear Burger equation. In this method, the trial space is generated by the multiquadric (MQ) RBF and the test space is generated by the compactly supported RBF. In the time discretization of the equation, the Taylor series expansion is used. This method is applied on some test experiments, and the numerical results have been compared with the exact solutions. The , , and root-mean-square (RMS) errors in the solutions show the efficiency and the accuracy of the method.

Solving partial differential equations by meshless methods using radial basis functions

In this paper, we will present some meshless methods for Partial differential equations, including meshless subdomain method, meshless Petrov–Galerkin method and meshless Schwarz method.

Convergence of meshless Petrov-Galerkin method using radial basis functions

In this paper, we provide a new meshless Petrov-Galerkin method, in which the trial space is generated by the global supported RBFs, the test space is generated by the compactly supported RBFs. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.