In recent years, there have been extensive efforts to find the numerical methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless method ...based on the weak form for interface problems. The proposed method combines the direct meshless local Petrov–Galerkin method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov–Galerkin method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov–Galerkin method is a newly developed modification of the meshless local Petrov–Galerkin method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov–Galerkin method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov–Galerkin method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson–Boltzmann and linear elasticity problems with two phases are studied. The proposed method is compared with analytical solution and the meshless local Petrov–Galerkin method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov–Galerkin method.
This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis ...functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.
One of the main problems of the conventional meshless methods in the time domain is their conditional stability. Up to now, some algorithms have been introduced to remove the stability condition, ...like alternating-direction-implicit (ADI) and locally one-dimensional (LOD) schemes. ADI meshless method has been investigated completely, but there is no report to exploit the full efficiency of the LOD meshless method. So, in this study, LOD scheme is employed to a meshless method for solving three-dimensional transient electromagnetic problems. The initial form of the proposed meshless method solves time-domain Maxwell's equations in three sub-steps, so it is called LOD3. LOD3 meshless method has first-order temporal accuracy. Moreover, LOD5 meshless method is introduced to upgrade temporal accuracy. Also, the results show that LOD5 meshless method is more accurate than ADI and LOD3 meshless methods. The accuracy of the proposed method is due to two main factors; Crank–Nicolson scheme and solving the equation in five sub-steps instead of one sub-step. Stability and accuracy of the proposed method are assessed through numerical experiment. Also, the unconditional stability of the proposed method is proved, analytically.
In this paper, a new coupled meshless method for 2D transient elastodynamic problems involving dynamic crack propagation is developed. The method is based on an efficient coupling between the finite ...point method (FPM) and a discretized form of Peridynamics. The solution domain is partitioned in three parts: one discretized by Peridynamics, one by FPM, and a transition part discretized by both methods where the switching between the two approaches is performed. The coupling adopts a local/nonlocal framework that benefits from the full advantages of both methods. The parts of domain where cracks either exist or are likely to propagate are described by Peridynamics; the remaining part of the domain is described by FPM that requires less computational effort. The capabilities of the proposed approach are demonstrated by means of the solution of dynamic problems including dynamic fracture as well as a ghost force test.
•A new coupled meshless method for elastodynamic problems as well as problems involving crack propagation is developed.•The proposed method is based on a suitable coupling between the meshless Finite Point method and a meshless Peridynamic method.•The coupling is done in a simple manner without introducing extra numerical artifacts.•The coupling technique is free of the presence of ghost forces.•The coupled method produces the solution of a Peridynamic-only model by using a much smaller number of nodes.
A strong-form boundary collocation method, the singular boundary method (SBM), is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures. The ...SBM is of several advantages including mathematically simple, easy-to-program, meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method. Due to the periodic behaviors of the structures, the SBM coefficient matrix can be represented as a block Toeplitz matrix. By employing three different fast Toeplitz-matrix solvers, the computational time and storage requirements are significantly reduced in the proposed SBM analysis. To demonstrate the effectiveness of the proposed SBM formulation for wave propagation analysis in periodic structures, several benchmark examples are presented and discussed The proposed SBM results are compared with the analytical solutions, the reference results and the COMSOL software.
In this paper a numerical approach based on the truly meshless methods is proposed to deal with the second-order two-space-dimensional telegraph equation. In the meshless local weak–strong (MLWS) ...method, our aim is to remove the background quadrature domains for integration as much as possible, and yet to obtain stable and accurate solution. The MLWS method is designed to combine the advantage of local weak and strong forms to avoid their shortcomings. In this method, the local Petrov–Galerkin weak form is applied only to the nodes on the Neumann boundary of the domain of the problem. The meshless collocation method, based on the strong form equation is applied to the interior nodes and the nodes on the Dirichlet boundary. To solve the telegraph equation using the MLWS method, the conventional moving least squares (MLS) approximation is exploited in order to interpolate the solution of the equation. A time stepping scheme is employed to approximate the time derivative. Another solution is also given by the meshless local Petrov-Galerkin (MLPG) method. The validity and efficiency of the two proposed methods are investigated and verified through several examples.
In this paper, two meshless methods, namely, a weak form Meshless Local Petrov Galerkin (MLPG) method, and Meshless Weak Strong (MWS) form method, obtained by combining MLPG with a strong form Radial ...Point Collocation Method (RPCM), are presented for simulation of advection-dispersion-reaction phenomena of the contaminants in the porous media. The first-order decay and sorption reactions are considered in this study. The Crank Nicolson scheme is applied for the time discretization. The weak form MLPG is a truly meshless and robust numerical technique, that can be applied to complex aquifer systems with derivative boundaries. However, in this method, the computational time is increased due to the integration, which is not essential for simple problems. Thus, the MLPG method is further coupled with a strong form RPCM with an aim of decreasing the background integration, by modelling only the nodes around the derivative boundaries using MLPG method and the other nodes by a direct RPCM which do not require integration. The proposed MWS model automatically converts into a complete RPCM model if there are no derivative boundaries. Thus, this model being both accurate and computationally efficient is suitable for simple and moderately complex aquifer systems and MLPG is the most stable and reliable method for modelling the most complex aquifer problems. Both the developed models are tested with available analytical solutions and applied for hypothetical case studies. The results prove the efficiency of the models and the applicability of each model is described in detail.
•Two new meshless models are proposed for simulation of reactive transport of contaminants undergoing sorption and decay•Meshless Local Petrov Galerkin (MLPG) weak form model is reliable for complex aquifers but has low computational efficiency•Meshless Weak Strong form model gives accurate solutions in lesser computational time
A reliable analysis of various groundwater problems requires accurate input of aquifer parameters. However, field measurement of such parameters is tedious and expensive. Inverse modelling by ...Simulation-Optimization (SO) resolves this limitation. In this study, the unknown transmissivity of confined aquifers is estimated using SO models. Three simulation models of strong, weak and hybrid categories of meshless methods, namely, Radial Point Collocation Method (RPCM), Meshless Local Petrov Galerkin (MLPG) and Meshless Weak Strong (MWS) form, are coupled with metaheuristic algorithms of Differential Evolution (DE), Particle Swarm Optimization (PSO) and Whale Optimization Algorithm (WOA), resulting in nine SO models. Five of these nine models are novel SO models and first-time application of WOA for groundwater flow parameter estimation. The application of models to heterogeneous hypothetical and complex field-type aquifers prove that solutions are similar to true transmissivities. This study provides an insight into the selection of suitable SO model based on available resources and requirements.
•Nine Simulation-Optimization (SO) models are presented for transmissivity estimation.•The flow simulations are based on meshless weak, strong and hybrid form methods.•Simulations are coupled with 3 metaheuristic algorithms and SO models are compared.•The relative advantages of each model and the choice of selection is discussed.
A reliable flow simulation is essential for better groundwater management. However, the complex flow behavior cannot be simulated by analytical methods and numerical models are often required. The ...Finite Difference and Finite Element Methods (FDM, FEM) are the most common and reliable numerical methods for groundwater studies. These methods require a background grid/mesh, which makes them complex. Thus, the meshless methods, which do not use a pre-defined grid/mesh, were developed. The meshless methods can be classified as strong and weak form methods. The strong form methods are simple and computationally efficient. The weak form methods are complex and require higher computational time, but are most suitable to derivative boundary problems. In this study, for the first time, groundwater flow behavior is studied by combining weak and strong form methods, with high accuracy in less computational time. The meshless weak strong form (MWS) model is developed by coupling the weak-form Meshless Local Petrov Galerkin (MLPG) method with the strong form Radial Point Collocation method (RPCM). The efficiency of the MWS model is verified using 1D and 2D hypothetical problems. Further, the MWS model is applied for a field problem and the results show the computational efficiency and accuracy of the model.
The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant ...system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.
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