Point Collocation Method Research Articles

Point collocation with mollified piecewise polynomial approximants for high‐order partial differential equations

AbstractThe solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell‐wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order of the piecewise functions and the smoothness of the mollifier. In this work, we exploit the high‐order and high‐smoothness properties of the mollified basis functions for solving PDEs through the point collocation method. The basis functions are evaluated at a set of collocation points in the domain. In addition, boundary conditions are imposed at a set of boundary collocation points distributed over the domain boundaries. To ensure the stability of the resulting linear system of equations, the number of collocation points is set larger than the total number of basis functions. The resulting linear system is overdetermined and is solved using the least square technique. The presented numerical examples confirm the convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. We study in particular the influence of the mollifier and the spatial distribution of the collocation points.

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.

Analysis of transverse free vibrations of rotating beams with variable cross-section using the point collocation method

In this article, a study of transverse free vibrations of rotating cantilever beams with variable cross-sectional area is carried out using the point collocation method. The study considers four thickness variation functions, including linear, parabolic, sinusoidal, and exponential types, for the beam’s cross-sectional area changes. The point collocation method uses Gauss-Legendre points to solve the equation of motion and determine the natural frequencies. To enhance the convergence rate, the initial guess function is based on the precise mode shapes of the stationary uniform beam. The effect of section functions and various rotational speeds on the structure’s natural frequencies is investigated. The outcomes of the investigation are compared with 3D finite element simulations and related published results. Through the analysis, the results show that the point collocation method is highly accurate, with a maximum error of 0.0025% compared to 3D finite element simulations. Moreover, the natural frequencies of the rotating cantilever beams decrease as the section functions become steeper. Finally, the increase in rotational speed causes the natural frequencies to increase in a non-linear manner, with the maximum increase rate occurring at a specific rotational speed.

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

In several engineering or physics problems, particularly those involving electromagnetic theory, thermal and radiation effects, acoustics, elasticity, and some fluid mechanics, it is not always easy or possible to find the analytical solution of integral equations that describe them. For this reason, numerical techniques are used. In this study, Point-collocation method was applied to linear and nonlinear, Volterra and Fredholm type integral equations and the performance and accuracy of the method was compared with several other methods that seem to be popular choices. As the base functions, a suitably chosen family of polynomials were employed. The convergence of the method was verified by increasing the number of polynomial base functions. The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations. Nonlinear problems take longer to calculate approximate solution coefficients than linear problems. Furthermore, it is necessary to use the real and smallest coefficients found in order to obtain a suitable approximate solution to these problems.

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

The attractiveness of the boundary element method—the reduction in the problem dimension by one—is lost when solving nonlinear solid mechanics problems. The point collocation method applied to strong-form differential equations is appealing because it is easy to implement. The method becomes inaccurate in the presence of traction boundary conditions, which are inevitable in solid mechanics. A judicious combination of the point collocation and the boundary integral formulation of Navier’s equation allows a pure boundary element method to be obtained for the solution of nonlinear elasticity problems. The potential of the approach is investigated in some simple examples considering isotropic and anisotropic material models in the total Lagrangian framework.

İNTEGRO-DİFERANSİYEL DENKLEMLERİN SAYISAL ÇÖZÜMÜNE UYGULANAN KOLLOKASYON YÖNTEMİ

Integro-differential equations are encountered in such fields of study as mechanics, physics, chemistry, biophysics, astronomy, economic theory, and population dynamics. In rare cases the solution methods for differential and/or integral equations can be generalized to integro-differential equations; but in general, numerical methods have to be applied. Recent years have seen the development of a large number of methods applicable to integro-differential equations. The present study aims to compare these newer methods with the classical method of point collocation, which is one of the weighted residual methods. The method was applied to test problems chosen from the literature, both linear and nonlinear integro-differential equations, and was seen to give good results.

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.

An adaptive divided-difference perturbation method for solving stochastic problems

Most stochastic problems are based on the independent and symmetrically distributed random variables. Using this principle, in this study, we propose an efficient calculation scheme to obtain the statistical moments based on high-order stochastic perturbation (SP) expansion. This solves the long-existing problem of the SP method, namely, the low calculation efficiency of statistical moments using high-order perturbation expansion, and the complexity of the calculation format. The scheme involves adopting the divided-difference method to approximate the partial derivative items of the SP method. The proposed method, which is similar to the collocation point method, only requires the deterministic calculation result for each node in the divided-difference method. In addition, as the adoption of high-order perturbation expansion is limited in multidimensional random problems due to the curse of dimensionality, an adaptive partial derivative items selection method is proposed in this paper to efficiently and reasonably ignore the expansion items that have an insignificant influence on statistical moments. This considerably improves the calculation efficiency of the statistical moments. Finally, the adaptive divided-difference perturbation (ADDP) method proposed in this paper is compared with the quasi-Monte Carlo method and the adaptive sparse grid methods using three numerical examples. The results prove that the ADDP method not only has high efficiency but also leads to significant improvements in the accuracy of statistical moments.

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.

A meshfree radial basis function method for simulation of multi‐dimensional conservation problems

AbstractMany computational fluid dynamics problems utilize finite volume frameworks for simulation, due to the simplifications provided by conservative formulation of the driving partial differential equations (PDEs). However, fluid dynamics applications can often involve temporal shifts in the domain structure—such as moving boundaries, or pore structure changes—requiring mesh adaptation throughout computation. These mesh adaptations often render classical numerical methods such as the finite volume method infeasible, due to their reliance on a well‐defined static mesh structure. This limitation has led to the development of a wide variety of meshless methods—techniques that can simulate PDEs without requiring a rigid connective structure between nodes. However, most meshless methods are typically based on finite element or finite difference formulations, and the limited number of meshless finite volume methods (MFVMs) either introduce a weak background mesh, or use weak‐form approximations that do not take full advantage of the strong conservative form of the driving equations. Addressing this gap within this study we outline a meshfree numerical scheme for simulation of partial different equations, based on strong‐form finite volume style formulations. Building upon the previously developed MFVM, this technique uses radial basis functions to interpolate the problem domain, and approximate fluxes in a disjoint finite volume scheme, removing reliance on a mesh structure. We present method derivation, including promising new techniques for enforcing boundary conditions in a meshless environment. Following this we discuss method accuracy and computational performance across a variety of problems in two and three dimensions. We then illustrate how this method may prove beneficial for applications in porous media modeling, and computational fluid dynamics. For completeness, we provide a sensitivity analysis of the method hyper‐parameters and investigate the conservative properties of the method. We also illustrate similarities of this approach to the widely used meshless point collocation methods. We close with a discussion of the strengths, limitations, and broader applicability of the technique.

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.

The effect of axial boundary conditions on breakout noise from finite cylindrical ducts

Ductborne noise within HVAC and exhaust systems may transfer into the surrounding environment through the walls of the duct. This breakout noise normally needs to be lowered in order to meet health and safety guidelines. This means that the ability of the duct walls to lower breakout noise needs to be predicted during the design stage of the duct systems. Significant work has been conducted into the prediction of breakout noise for infinite length ducts, however there is little to be found on finite length ducts. In particular there are very few studies on the difference between finite and infinite length duct breakout, especially where the noise source lies within the internal fluid. The aim of this work is therefore to investigate the difference in breakout noise between finite length and equivalent infinite length ducts when excited by an internal noise source. This is performed through numerical experiments using the semi analytical finite element method which enables the equations of elasticity for the duct wall, as well as a surrounding fluid, to be accommodated. Axial continuity equations at each end of a finite length elastic duct are then enforced through the point collocation method. It is observed that high sound power levels are emitted at axial resonances of the finite length duct. In the low frequency region these resonances have a narrow bandwidth and are expected to be of little practical significance. However, above the critical frequency the resonance bandwidth increases and this is observed to significantly lower the transverse transmission loss of the duct wall when compared to an infinite length duct. This phenomenon is observed for both clamped and simply supported ducts, as well as for two different internal sound sources. It is concluded that breakout noise from axial resonances in finite length ducts should be examined in design calculations in order to avoid excessive breakout noise.

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.

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL (Low-density Lipoprotein), HDL (High-density Lipoprotein) and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the first-order optimality conditions. Numerical results are considered to illustrate the efficiency of methods.

An immersed boundary vector potential‐vorticity meshless solver of the incompressible Navier–Stokes equation

AbstractWe present a strong form meshless solver for numerical solution of the nonstationary, incompressible, viscous Navier–Stokes equations in two (2D) and three dimensions (3D). We solve the flow equations in their stream function‐vorticity (in 2D) and vector potential‐vorticity (in 3D) formulation, by extending to 3D flows the boundary condition‐enforced immersed boundary method, originally introduced in the literature for 2D problems. We use a Cartesian grid, uniform or locally refined, to discretize the spatial domain. We apply an explicit time integration scheme to update the transient vorticity equations, and we solve the Poisson type equation for the stream function or vector potential field using the meshless point collocation method. Spatial derivatives of the unknown field functions are computed using the discretization‐corrected particle strength exchange method. We verify the accuracy of the proposed numerical scheme through commonly used benchmark and example problems. Excellent agreement with the data from the literature was achieved. The proposed method was shown to be very efficient, having relatively large critical time steps.

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 meshfree point collocation method for elliptic interface problems

We present a meshfree generalized finite difference method for solving Poisson’s equation with a diffusion coefficient that contains jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity; and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation for enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch from the strong form to the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases of varying complexity and jump magnitude on point clouds with nodes that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity.

Uniform Torsion Analysis Of Composite Shafts Using Point Collocation Method Based On Pascal Polynomials

Torsion bars are important elements in the marine structure applications. The usage of the composite torsion bars has increased recently because it takes advantage of the different material property of each constituent. The current research focuses mainly on the uniform warping analysis of composite torsion shafts by using the point collocation method (PCM). The PCM is based on Pascal's triangle polynomial expansion and coupled with subdomain technique that is indispensable for the analysis of composite bars. In addition to satisfying the field equation (Laplace equation for warping function) and traction-free boundary condition for each component, the continuity conditions in both warping functions and tractions need to be imposed along the interfaces of all components. In this paper, multiple-scale factors are introduced into the pol-ynomial series to improve the numerical stability. The current scheme is easy to be implemented, and the algebraic equations for warping functions can be simply established. This nu-merical method can effectively calculate the structural re-sponses of the composite shafts. The numerical examples of this study include the composite shafts having elliptical and rectangular cross sections. The structural responses such as warping functions, shear stresses and torsional rigidity are reported. The PCM solutions are compared with the exact solutions and with the finite element solutions obtained from COMSOL. The results show that this simple numerical method can effectively achieve the solutions with a high de-gree of accuracy.

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.

In Situ Bioremediation of Phenol through Confined Aquifer Using Mesh-Free Point Collocation Method with Radial Basis Function

In Situ Bioremediation of Phenol through Confined Aquifer Using Mesh-Free Point Collocation Method with Radial Basis Function