Radial Point Collocation Method Research Articles

Aquifer flow parameter estimation using coupled meshless methods and metaheuristic algorithms

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.

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.

Estimation of contamination release histories using meshless radial point collocation method and multiverse optimization

Abstract Contaminants in groundwater may enter through various sources which are required to be identified for informed decision-making regarding remediation. In early stages of contamination of an aquifer, the sources of contamination are generally unknown. To estimate the unknown release histories at potential sources, the governing equations of contaminant transport are solved backwards in time which renders the inverse problem as ill-posed. Furthermore, due to measurement or operational errors, observed breakthrough curves can be noisy which can induce uncertainty. Simulation-optimization (SO) techniques are generally used for solving the inverse model. Nevertheless, swarm population-based optimization algorithms may get trapped in local optima in the nonlinear search space related to the problem. In this study, SO model based on meshless radial point collocation method (RPCM) and multiverse optimizer (MVO) is proposed. MVO implements gradual transformation from global exploration to local exploitation phase which helps in better estimation of unknowns. The performance of RPCM-MVO-SO model is compared with two other SO models developed by combining RPCM with grey wolf optimizer and particle swarm optimization in two case studies. The proposed model performs better compared to the other two SO models considered which suggests the efficacy of the model for release history estimation in groundwater.

On the convergence of finite integration method for system of ordinary differential equations

The current work investigates basic theory of finite integration method for first-order system of ordinary differential equations. We firstly investigate the error analysis of generalized finite integration methods which are constructed by ordinary linear approach. Then, the basic concepts of the error bounded theorems are discussed for a spectral meshless method derived from quadrature rule producing by radial point collocation method. By applying the finite integration methods for the first-order system of ordinary differential equation, we compare the accuracy residual error of the presented methods. The convergence of finite integration methods is proposed to confirm the standard criteria with theoretically accurate results. Finally, we extend the results for n-order system of ordinary differential equations.

Simulation of reactive transport in porous media using Meshless Local Petrov Galerkin (MLPG) and combination of Meshless Weak and Strong (MWS) form methods

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.

Coupled flow and transport simulation involving rate-limited adsorption in highly heterogeneous unconfined aquifers using a local strong form meshless method

Adsorption plays a vital role in contaminant transport in groundwater. Generally, a local equilibrium assumption is used in which the retardation factor is derived from aquifer material and contaminant type. However, for problems involving faster seepage velocities, this assumption becomes invalid and the mass transfer between mobile and immobile or, sorbed phases of contaminant becomes rate-limited. Traditionally, finite difference method (FDM) models are employed for contaminant transport simulation with rate-limited adsorption (RLA). Nonetheless, due to the presence of a grid, these models show instabilities, especially for advection dominant problems. In the present study, a meshless Radial Point Collocation Method (RPCM-RLA) is proposed for contaminant transport simulation with rate-limited adsorption. The proposed model is verified with semi-analytical solutions and its efficacy is assessed with the popularly used FDM-based Reactive Transport in 3 Dimensions (RT3D) model. The RPCM-RLA model is observed to provide stable and accurate solutions even for advection dominant problems whereas RT3D model simulations show excessive numerical dispersion. Furthermore, the effectiveness of RPCM-RLA model is demonstrated in two case studies concerning heterogeneous unconfined aquifers by comparing it with RT3D. Both model simulations agree consistently with maximum values of normalized mean absolute error (NMAE) being 0.0055 and 0.031 for case studies 1 and 2 respectively, with better performance of RPCM-RLA model.

A meshless weak strong form method for the groundwater flow simulation in an unconfined aquifer

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.

A Meshless Local Radial Point Collocation Method for Simulating the Time-Fractional Convection-Diffusion Equations on Surfaces

The time-fractional problem is a class of important models to represent the real world. It is an open problem to study how the fractional operator acts on the surface. In this work, we present and analyze a meshless local radial point collocation method for numerically solving time-fractional convection-diffusion equations on closed surfaces embedded in [Formula: see text]. The second-order shifted Grünwald scheme is applied in time discretization. All computations use only extrinsic coordinates to avoid coordinate distortions and singularities. Moreover, the stability and convergence of the method are proven by the energy estimate. Numerical experiments are provided to support the convergence analysis and numerical performance of the proposed method.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

Meshfree radial point collocation-based coupled flow and transport model for simulation of multispecies linked first order reactions

Mesh/grid-based methods such as the Finite element method (FEM) and Finite difference method (FDM) are extensively used for simulation of contaminant species transport in groundwater. However, these methods in their standard form are susceptible to instabilities such as oscillations and numerical dispersion in the solution. Further, due to the use of mesh/grid, these methods are also computationally expensive. In this study, a meshfree method named Radial point collocation method (RPCM) is demonstrated for simulating reactive transport involving linked first-order reactions in a coupled flow environment. Multiquadrics radial basis functions (MQ-RBF) are used for approximation of the state variables. The model uses local support domains which produce sparse matrix systems that efficiently deal with the ill-conditioning problem which generally occurs while using global support domains. The model developed is verified against the semi-analytical solutions and its performance is compared with FEM for case studies. It is observed that the model provides accurate solutions for the problems considered and also handles advective flow conditions better than FEM. The proposed model successfully simulates the fate of contaminants in linked chain reactions.

Modeling of transport of first-order reaction networks in porous media using meshfree radial point collocation method

First-order reaction networks are often encountered in groundwater pollution problems. The members of the network transform/degrade from one species to another. In this paper a strong form meshfree numerical method named radial point collocation method (RPCM) is introduced to model such reactions. In the proposed method, the entire domain is decomposed into a number of overlapping local support domains surrounding each node. The concentrations of species at each node and its directional derivatives are approximated using multiquadrics radial basis functions (MQ-RBF) within the local support domain. 1D models are verified against existing analytical solutions. Further, the 2D model simulations are compared with widely used RT3D (Reactive Transport in 3 Dimensions) for a benchmark problem. Sensitivity analysis is also performed for model parameters. The proposed model solves species-wise governing equations and boundary conditions without operator splitting, unlike the popular grid/mesh-based models. The RPCM-based models are found to be very effective in modeling first-order reaction network problems.

Simulation of reactive transport in porous media using radial point collocation method

In this paper, a meshfree Radial Point Collocation method (RPCM) for modelling of reactive transport involving first-order decay and adsorption is presented. Unlike conventional grid-based methods such as Finite Element (FEM) and Finite Difference Methods (FDM), it does not require nodal connectivity information for domain representation. The proposed method does not necessitate operator splitting for accommodating reactions as it is done in FEM and FDM. The model domain is divided into intersecting circular shaped local support domains. This technique solves the ill-conditioning problem associated with globally supported system and allows use of more number of nodes which increases the accuracy of models. The multi-quadrics radial basis function (MQ-RBF) is used for the approximation/interpolation of solute concentration values in the local support domain. 1D and 2D models are developed and verified against benchmark analytical solutions. Sensitivity analyses are performed for shape parameters of the MQ-RBF, support domain size and nodal density. Two case studies are presented with regular and irregular shaped domains and results are compared with FEM. The results demonstrate advantages of the proposed model over grid-based methods while handling adsorption and first-order decay reactions. This study shows that the proposed model is effective for simulating reactive transport problems in porous media.

Optimal remediation design and simulation of groundwater flow coupled to contaminant transport using genetic algorithm and radial point collocation method (RPCM)

The simulation-optimisation models of groundwater and contaminant transport can be a powerful tool in the management of groundwater resources and remediation design. In this study, using Multiquadratic Radial Basis Function (MRBF) a coupled groundwater flow and reactive transport of contaminant and oxidant was developed in the framework of the Meshfree method. The parameter analysis has determined the optimum shape parameter (0.97), and the results of the model were compared with a physical sandbox model which were in good agreement. The genetic algorithm approach was used to find the optimum design of the remediation using permanganate as an oxidant. To find the optimum design we considered two objectives and two constraints. The results revealed that the breakthrough of contaminant to the downstream area of interest and the concentration of the contaminant in this area is reduced significantly with optimisation.

A comparative study in aquifer parameter estimation using MFree point collocation method with evolutionary algorithms

Abstract In this study, we present a comparative assessment of simulation-optimization (S-O) models to estimate aquifer parameters such as transmissivity, longitudinal dispersivity, and transverse dispersivity. The groundwater flow and contaminant transport processes are simulated using the mesh-free radial basis point collocation method (RPCM). Four different S-O models are developed by combining the RPCM model separately with genetic algorithm (GA), differential evolution (DE), cat swarm optimization (CSO), and particle swarm optimization (PSO). The objective of the S-O model is to minimize a composite objective function with transmissivity, longitudinal dispersivity, and transverse dispersivity as decision variables. Hydraulic head and contaminant concentration at observation points are the state variables. The S-O models are used to estimate aquifer parameters of a confined aquifer with nine zones. It is found that RPCM-based DE, CSO, and PSO models are more accurate in estimating aquifer parameters than RPCM-GA. However, for noisy observed data, the RPCM-CSO model outperforms other models. The efficiency of the RPCM-CSO model over other models is further established by performing reliability analysis to the noisy observed data set. The comparative study reflects the efficacy of CSO over GA, DE, and PSO.

Solving acoustic scattering problem by the meshless method based on radial basis function

In this paper, radial point collocation method (RPCM) is introduced to solve the acoustic scattering problem. This is a mathematically simple, easy-to-program and truly meshless method, which has been successfully applied to solve the solid mechanics and convection diffusion problems. However, application of this method to investigate acoustic problems, in particle the acoustic scattering problem is relatively new. The main advantage of this method is its mathematically simple, easy to program, and truly meshless. A Hermite-type interpolation method is employed to improve the solution accuracy while the Neumann boundary conditions exist. In addition, acoustic scattering problem is a typical unbounded domain problem, in order to solve it with RPCM, the domain is truncated to a finite region and an artificial boundary condition (ABC) is imposed. Finally, numerical example is presented to validate the accuracy and effectiveness of RPCM. In the future, the extension of RPCM to more complex and practical problems, especially the three-dimensional situations need to be investigated in more detail.

Identification of Groundwater Contamination Sources Using Meshfree RPCM Simulation and Particle Swarm Optimization

In this study, a simulation-optimization (SO) model is presented by coupling a meshfree based simulator using radial point collocation method (RPCM) and particle swarm optimization (PSO) as optimizer to identify the unknown groundwater contaminant sources from the measured/simulated contaminant concentration data in the aquifer. To demonstrate the approach, two case studies have been presented. The first example is a hypothetical case which simulates the contaminant releases from several disposal sites in an aquifer during four years release period. The second case considered is a field study where leaching of contaminant, during their storage, from disposal sites at several locations in the aquifer leads to contamination of the groundwater. The goal in both cases was to reconstruct the contaminant release history from the disposal sites and their magnitudes from the given historical concentration data at a few observation wells in the aquifer. It was observed that the source identification model could reconstruct the release histories from the waste disposal sites in both the cases accurately. This study demonstrated that PSO based optimization model with a meshfree flow and transport simulator can be effectively used for groundwater contaminant source identification problems.

Coupled groundwater flow and contaminant transport simulation in a confined aquifer using meshfree radial point collocation method (RPCM)

In this study, a meshfree radial point collocation method is used to model the contaminant transport through confined aquifer. The discretization of the governing equations is done by a point collocation method and radial basis functions (RBF) are used as the interpolation function. For comparative study, two widely used radial basis functions namely multi-quadrics and exponential RBF are used. A local circular support domain is employed to construct the shape functions. In the model, no information on nodal inter-relationship is required for shape function construction except the nodal coordinates, unlike in finite-difference (FDM) or finite-element (FEM) based methods. The developed model is validated through benchmark problems in one and two dimensions. Further, application of the model for advective transport with high Peclet number has been studied and the model has been found to be effective in handling the instability of high Peclet problems. For the field problem considered, the results obtained from the model have been compared with the FEM solution and was found to be satisfactory. This method is relatively easy to implement and offers better accuracy with acceptable computational time. Considering the significant advantages offered by this method, it can serve as a good alternative to the conventional methods.

A nth-order shear deformation theory for the free vibration analysis on the isotropic plates

In the present paper, a nth-order shear deformation theory is used to perform the free vibration analysis of the isotropic plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third order theory can be considered as a special case of present nth-order theory (n=3). The governing equations and boundary conditions are derived by the principle of virtual work. The governing differential equations of the isotropic plates are solved by the meshless radial point collocation method based on the thin plate spline radial basis function. The effectiveness of the present theory is demonstrated by applying it to free vibration problem of the square and circular isotropic plate.

Meshless global radial point collocation method for three-dimensional partial differential equations

This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.

A meshless local radial point collocation method for free vibration analysis of laminated composite plates

In this paper, a meshless local radial point collocation method based on multiquadric radial basis function is proposed to analyze the free vibration of laminated composite plates. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, and material properties are computed by present method. The choice of shape parameter, effect of dimensionless sizes of the support domain on accuracy, convergence characteristics are studied by several numerical examples. The results are compared with available published results which demonstrate the accuracy and efficiency of present method.