Point Collocation Method Research Articles

Application of fixed point-collocation method for solving an optimal control problem of a parabolic–hyperbolic free boundary problem modeling the growth of tumor with drug application

In this paper, employing a fixed point-collocation method, we solve an optimal control problem for a model of tumor growth with drug application. This model is a free boundary problem and consists of five time-dependent partial differential equations including three different first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the mentioned optimal control problem, the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells. In this study, applying the fixed point method, we construct a sequence converging to the solution of the optimal control problem. In each step of the fixed point iteration, the problem changes to a linear one and the parabolic equations are solved using the collocation method. The stability of the method is also proved. Some examples are considered to illustrate the efficiency of method.

Analysis of Piezoelectric Solids using Finite Element Method

Piezoelectric materials are extensively used in smart structures as sensors and actuators. In this paper, static analysis of three piezoelectric solids is done using general-purpose finite element software, Abaqus. The simulation results from Abaqus are compared with the results obtained using numerical methods like Boundary Element Method (BEM) and meshless point collocation method (PCM). The BEM and PCM are cumbersome for complex shape and complicated boundary conditions. This paper shows that the software Abaqus can be used to solve the governing equations of piezoelectric solids in a much simpler and faster way than the BEM and PCM.

Moving boundary analysis in heat conduction with multilayer composites by finite block method

An inverse reconstruction investigation is presented to determine the inner boundary location (corrosion points) for the heat transfer in composite walls from measurement data on exterior boundary. Finite Block Method (FBM) is utilized in this paper to deal with transient heat problems across the multilayered composite walls. Starting from one-dimensional problems, Lagrange interpolation with equally spaced nodes is applied to create first order differential matrices and thereafter the higher order differential matrices are obtained. Then combining with mapping technique, physical domain is mapped into a normalized domain for two-dimensional or three-dimensional problems with 8 seeds or 20 seeds respectively. Both time-spatial approach and Laplace transform technique with Durbin's inversion method are employed in the simulating procedure. In addition, roots of Chebyshev polynomial of first kind are considered in FBM for the first time, which can improve the degree of convergence significantly. Three numerical examples are presented to validate the accuracy of FBM. Comparisons between Finite Element Method (FEM), FBM and Point Collocation Method (PCM) are demonstrated respectively. Numerical observation indicates that FBM has much higher degree of accuracy even with few collocation points.

In-situ bioremediation of groundwater using a meshfree model and particle swarm optimization

AbstractGroundwater contamination due to contaminants like trichloroethylene (TCE), tetrachloroethylene, dichloroethylene, phenol, etc., is an alarming concern for most of the manufacturing areas. It is important to identify the type of pollutant, concentration, location, and direction of the contaminant plume for groundwater remediation. Bioremediation has been identified as one of the important remediation techniques for these types of contaminants. Bioremediation modeling comprises solutions to biodegradation equations and fixing the time of remediation and locating the oxygen injection wells. In this study, a simulation-optimization (S/O) model based on the coupled meshfree point collocation method (MFree-PCM) and particle swarm optimization (PSO) is proposed for in-situ bioremediation design. The in-situ bioremediation process of groundwater contamination is explored using the developed PCM-BIO-PSO multi-objective model with different strategies of minimization of cost, number of wells and time of remediation. The proposed model can be effectively used for the in-situ bioremediation design of contaminated sites.

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.

Study on the Single Scattering of Elastic Waves by a Cylindrical Fiber with a Partially Imperfect Bonding Using the Collocation Point Method

The single scattering of P‐ and SV‐waves by a cylindrical fiber with a partially imperfect bonding to the surrounding matrix is investigated, which benefits the characterization of the behavior of elastic waves in composite materials. The imperfect interface is modelled by the spring model. To solve the corresponding single scattering problem, a collocation point (CP) method is introduced. Based on this method, influence of various aspects of the imperfect interface on the scattering of P‐ and SV‐waves is studied. Results indicate that (i) the total scattering cross section (SCS) is almost symmetric about the axis α = π/2 with respect to the location (α) of the imperfect interface, (ii) imperfect interfaces located at α = 0 and α = π highly reduce the total SCS under a P‐wave incidence and imperfect interfaces located at α = π/2 reduce the total SCS most significantly under SV‐incidence, and (iii) under a P‐wave incidence the SCS has a high sensitivity to the bonding level of imperfect interfaces when α is small, while it becomes more sensitive to the bonding level when α is larger under SV‐wave incidence.

Orthotropy in Annular fins: A Collocation based solution

The effect of orthotropy in 2D temperature distribution in annular fins is reported using collocation method. Point Collocation Method (PCM) gives results for insulated tip boundary condition problem along with contact resistance at the base. The problem is solved assuming the non-dimensional 4th order polynomial temperature distribution and forcing the residue to zero at Gauss quadrature points. The fin material is assumed to be aluminium (Kr =200w/m k). The fin is orthotropic in nature and thermal conductivity effect has been governed by thermal conductivity ratio (K). The analysis has been done by considering different coefficients of convective heat transfer at fin base (hc), surface (hs) and tip (he) and is reported for three different aspect ratios (α) of fin. The results presented demonstrate the viability for design and optimization of fins of different sizes under a range of convection conditions, applied to many practical applications encountered in engineering industry.

Scattering Attenuation of SH‐Waves in Fiber‐Reinforced Composites with Partial Imperfectly Bonded Interfaces

The influence of partial imperfectly bonded interfaces between each fiber and the matrix on the scattering attenuation of coherent SH‐waves in fiber‐reinforced composites is investigated. The imperfection of interfaces is modelled using the spring model, in which the level of imperfection is characterized by a parameter K, which is called the stiffness of the imperfect interfaces. First, the single scattering of SH‐waves by a cylindrical fiber with such a partial imperfectly bonded interface to the matrix is formulated and subsequently solved using the collocation point (CP) method. Later on, based on the analysis of the corresponding single scattering problem, effects of several parameters (i.e., frequency of the incident wave, level of imperfection of interfaces, and width of the imperfect interfaces) on the far‐field scattering magnitude, scattering cross section, and scattering attenuation coefficients of coherent SH‐waves are shown graphically. The potential application of the current results to nondestructive evaluation of interfaces in composites is also discussed.

Vibroacoustic modeling of an elastic beam in low subsonic flows with mean velocities

Abstract The paper presents a vibroacoustic model of an elastic beam submerged in uniform low subsonic flows. The governing equations of the elastic beam and acoustic fields are reformulated by means of a coordinates transformation. The convective terms in boundary conditions of the model are considered to achieve the decoupled integral equations. The dynamic responses and radiated acoustic fields of the beam are computed numerically by using the collocation point method. In the numerical examples, the sound waves radiated from the beam are shifted to the moving direction of fluids. Meanwhile, new crests of sound waves are formed in the upstream direction of the sound source with the Mach number increasing. In addition, the numerical results are independently evaluated by power flows of the system.

Effective interfacial conditions for the Stokes flow of a fluid on periodically rough surfaces

This work is concerned with the Stokes flow of a fluid through the channel between two parallel solid walls, of which one is smooth and homogeneous and the other is periodically rough and heterogeneous. The main purpose of this work is to homogenize the resulting periodically rough solid/fluid interface so as to replace it by an equivalent smooth homogeneous solid/fluid interface characterized by appropriate effective slip lengths. To achieve this objective, a semi-analytic approach is elaborated by carrying out the Fourier expansions of the velocity and pressure fields in the plane normal to the thickness direction of the channel and by resorting to the point collocation method. In comparison with the relevant approaches reported in the literature, the present one has the advantage of being valid for the general case where the distance between the two parallel walls is arbitrary. With the help of the elaborated approach, the effective slip lengths are determined for a variety of periodically rough surface microstructures including, for example, longitudinal grooves, transverse grooves, chessboard texture and the Hashin–Shtrikman’s nested circles. The results obtained by the proposed semi-analytical approach are compared systematically with the ones provided by the finite element method and discussed in the light of some available bounds or analytical results. It follows from these comparisons and discussions that our semi-analytical approach is particularly efficient and accurate.

Application of Results of Tests in Developing a Two-Dimensional Model for Soil–Steel Railway Bridges

This paper presents an analysis of the possibilities for the application of a two-dimensional model based on a circumferential strip of soil–steel structures that used results of measurements on railway soil–steel bridges. The distribution of a live load in the model was based on the so-called collocation point method. The method assumed conformity of results coming from a real test and those obtained in the two-dimensional model at a certain point of a structure. As a conformity criterion, deformation or internal forces (e.g., bending moment, thrust force, and stress) was assumed. Two locomotives moving above the structures were used to obtain influence lines of the collocation point. Examples of influence lines are presented; these also could be used for determination of stiffness of soil–steel bridges. The main advantage of this approach is the ability to evaluate the effects of any moving loads applied on existing bridges according to findings of measurements of deflections and development of deflection influence lines.

Post buckling analysis for composite plate by finite block Petrov-Galerkin method

Abstract Based on the Lagrange series interpolation, the Finite Block Petrov-Galerkin Method (FBPGM) has been developed to the non-linear analysis of Reissner plate theory with Functionally Graded Materials (FGMs). By introducing the mapping technique, a quadratic type block is mapped into a normalized domain with 8 seeds. Then the domain nodes distribution can be determined by using shape functions. The load incremental technique and displacement control scheme are adopted to solve nonlinear problems including large deformation and post buckling of plates under pressure load and in-plane compressive forces. Numerical examples are given to demonstrate the efficiency and accuracy of proposed method in comparing with the Boundary Element Method (BEM), point collocation method (PCM) and Finite Element Method (FEM).

A collocated isogeometric finite element method based on Gauss–Lobatto Lagrange extraction of splines

Generalizing the concept of Bézier extraction, we introduce an extraction operator that links C0 Gauss–Lobatto Lagrange functions with smooth splines. This opens the door for collocated isogeometric analysis that combines the accuracy of the Galerkin method with collocation-type formation and assembly procedures. We present the key ingredients of the technology, i.e. integration by parts and the weighted residual form, the interaction of Gauss–Lobatto Lagrange extraction with Gauss–Lobatto quadrature, and symmetrization with the ultra-weak formulation. We compare the new method with standard isogeometric Galerkin and isogeometric point-collocation methods for spline discretizations in three dimensions.

A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation

In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O(ź3źź), 1<ź < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method.

A numerical approximation method for nonlinear dynamic systems based on radial basis functions

The radial basis functions have the advantages of simple forms and isotropy. A new numerical method for solving the initial-value problems of nonlinear dynamic systems was constructed through combination of the idea of the radial basis function approximation and the weighted residual collocation point method. The advantages and disadvantages of several methods for the numerical solution of nonlinear dynamic systems were analyzed. Some practical numerical examples were given to compare the proposed method with the existing methods. The results show that the present method is easily applicable with good convergence and high accuracy.

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.

Simulation of Seawater Intrusion in Coastal Confined Aquifer Using a Point Collocation Method Based Meshfree Model

Seawater intrusion caused by groundwater over-exploitation from coastal aquifers poses a severe problem in many regions. Formulation of proper pumping strategy using a simulation model can assure sustainable supply of fresh water from the coastal aquifers. The focus of the present study is on the development of a numerical model based on Meshfree (MFree) method to study the seawater intrusion problem. For the simulation of seawater intrusion problem, widely used models are based on Finite Difference (FDM) and Finite Element (FEM) Methods, which demand well defined grids/meshes and considerable pre-processing efforts. Here, MFree Point Collocation Method (PCM) based on the Radial Basis Function (RBF) is proposed for the simulation. Diffusive interface approach with density-dependent dispersion and solution of flow and solute transport is adopted. These equations are solved using PCM with appropriate boundary conditions. The developed model has been verified with Henry’s problem, and found to be satisfactory. Further the model has been applied to another established problem and an attempt is made to examine the influence of important system parameters including pumping and recharge on the seawater intrusion. The PCM based MFree model is found computationally efficient as preprocessing is avoided when compared to other numerical methods.

Meshless point collocation methodによるコンクリートの破壊解析

Meshless point collocation methodによるコンクリートの破壊解析

Nonlinear dynamic response and damage analysis for functionally graded metal shallow spherical shell under low-velocity impact

The nonlinear dynamic response and damage evolution of functionally graded shallow spherical shell under low-velocity impact are investigated in this work. Basing on continuum damage theory, a damage constitutive relation is established for functionally graded material and the Kachanov damage evolution law is adopted to predict the damage propagation in the structure. A modified contact model suitable for non-homogenous material (functionally graded material) is applied to model the contact force in impacting process. The laminated modeling method is adopted to model the functionally graded shell with varying material constants along the thickness by dividing the shell to N plies with the constant material properties for each ply. With the established damage constitutive relations and nonlinear geometric relations of FGM shallow spherical shell with elastic modulus varying as a power-law function, the nonlinear motion equations of FGM shallow spherical under low-velocity impact have been obtained in the term of displacement functions. The problems are solved by using the orthogonal collocation point method, the Newmark method and the iterative method synthetically. Some numerical examples are carried out to validate present impacting model and the calculating methods, and parametrical analysis are presented to discuss the effects of the material properties, the geometrical size and impacting velocity on damage state and dynamic response of the structure when under low-velocity impact.

Finite integration method for solving multi-dimensional partial differential equations

Based on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial Basis Function (RBF) interpolation (uniform/random distributions of nodes) to higher order integration matrices. Using standard time integration techniques, such as Laplace transform, we have shown that the FIM is capable for solving time-dependent partial differential equations. Illustrative numerical examples are given in two-dimension to compare the FIM (FIM-OLA and FIM-RBF) with the finite difference method and point collocation method to demonstrate its superior accuracy and efficiency.