Order Collocation Method Research Articles

Numerical Solution of Burgers–Huxley Equation Using a Higher Order Collocation Method

In this paper, the collocation method with cubic B-spline as basis function has been successfully applied to numerically solve the Burgers–Huxley equation. This equation illustrates a model for describing the interaction between reaction mechanisms, convection effects, and diffusion transport. Quasi-linearization has been employed to deal with the nonlinearity of equations. The Crank–Nicolson implicit scheme is used for discretization of the equation and the resulting system turned out to be semi-implicit. The stability of the method is discussed using Fourier series analysis (von Neumann method), and it has been concluded that the method is unconditionally stable. Various numerical experiments have been performed to demonstrate the authenticity of the scheme. We have found that the computed numerical solutions are in good agreement with the exact solutions and are competent with those available in the literature. Accuracy and minimal computational efforts are the key features of the proposed method.

THERMAL CONVECTION OF MAGNETO-HYDRODYNAMIC MICROPOLAR FLUID FLOW OVER A POROUS NON-LINEAR STRETCHING SHEET WITH THERMAL RADIATION

The thermal convection of a magneto-hydrodynamic (MHD) micropolar fluid flow in a porous nonlinear media stretching sheet with thermal radiation is investigated in this paper. The governing equations are stated in non-linear form, and the solution of non-linear equations is obtained by translating them in ODE using an appropriate similarity transformation. The influence of the power index (n), material parameter (A), magnetic field parameter (M), porosity parameter (Kp), and Prandtl number on fluid velocity, micro-rotation, temperature distribution, and skin friction coefficient has been investigated. Furthermore, the physical consequences of the aforementioned factors on flow are explored. The MAT LAB software bvp4c (boundary value problem fourth order collocation method) is used to predict the graphs between various quantities.

Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method

New analytical solutions for the static deflection of anisotropic composite beams resting on variable stiffness elastic foundations are obtained by the means of the Homotopy Analysis Method (HAM). The method provides a closed-form series solution for the problem described by a non-homogeneous system of coupled ordinary differential equations with constant coefficients and one variable coefficient reflecting variable stiffness elastic foundation. Analytical solutions are obtained based on two different algorithms, namely conventional HAM and iterative HAM (iHAM). To investigate the computational efficiency and convergence of HAM solutions, the preliminary studies are performed for a composite beam without elastic foundation under the action of transverse uniformly distributed loads considering three different types of stacking sequence which provide different levels and types of anisotropy. It is shown that applying the iterative approach results in better convergence of the solution compared with conventional HAM for the same level of accuracy. Then, analytical solutions are developed for composite beams on elastic foundations. New analytical results based on HAM are presented for the static deflection of composite beams resting on variable stiffness elastic foundations. Results are compared to those reported in the literature and those obtained by the Chebyshev Collocation Method in order to verify the validity and accuracy of the method. Numerical experiments reveal the accuracy and efficiency of the Homotopy Analysis Method in static beam problems.

The effects of Coriolis force and radial stretch on the convective instability characteristics of the Bödewadt flow

The Bödewadt boundary-layer flow is induced by the rotation of a viscous fluid rotating with a constant angular velocity over a stationary disk. In this paper, the Bödewadt boundary-layer flow has been studied in the presence of the Coriolis force to observe the effect of radial stretch of the lower disk on the flow. For the first time in the literature, a numerical investigation of the effects of both stretching mechanism and the Coriolis force on the flow behaviour and on the convective instability characteristics of the above flow has been carried out. In this paper, the Kármán similarity transformations have been considered in order to convert the system of PDEs representing the momentum equations of the flow into a system of highly non-linear coupled ODEs and solved numerically to obtain the velocity profiles of the Bödewadt flow. Then, a convective instability analysis has been performed by using the Chebyshev collocation method in order to obtain the neutral curves. From the neutral curves it is observed that radial stretch has a globally stabilising effect on both the inviscid Type-I and the viscous Type-II instability modes. This underlying physical phenomena has been verified by performing an energy analysis of the flow. The results obtained excellently supports the previous works and will be prominently treated as a benchmark for our future studies.

Coordination of construction manipulation robotic system using UAV

Drone technology has demonstrated to be of great contribution to the construction field. Tasks such as photogrammetry, plot surveillance and assets inspections are successfully automated with the aerial robotic technology with some level of contribution of a human operator. In this paper we analyse the possibility to achieve fully automated construction task using dispersed robotic system: a drone and mobile robotic crane. The role of the drone is to study the surroundings and generate a desired trajectory for the crane boom arm in order to lay-in pergola blades. The paper focuses on the layers of the dispersed cyber-physical system allowing to achieve such complex task and optimize the trajectory of the crane boom arm by resolving two-dimensional Chebyshev-Gauss collocation method in order to have minimum-jerk path.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal diffusion are asymptotically compatible. We verify these findings through extensive numerical experiments.

Variance-Based Robust Optimization of a Permanent Magnet Synchronous Machine

This paper focuses on the application of the variance-based global sensitivity analysis for a topology derivative method in order to solve a stochastic nonlinear time-dependent magnetoquasistatic interface problem. To illustrate the approach a permanent magnet synchronous machine has been considered. Our key objective is to provide a robust design of the rotor poles and of the tooth base in a stator for the reduction of the torque ripple and electromagnetic losses, while taking material uncertainties into account. Input variations of material parameters are modeled using the polynomial chaos expansion technique, which is incorporated into the stochastic collocation method in order to provide a response surface model. Additionally, we can benefit from the variance-based sensitivity analysis. This allows us to reduce the dimensionality of the stochastic optimization problems, described by the random-dependent cost functional. Finally, to validate our approach, we provide the two-dimensional simulations and analysis, which confirm the usefulness of the proposed method and yield a novel topology of a permanent magnet synchronous machine.

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where $$n+1$$ denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order $$\mathcal{O}(n^{-m}\log n)$$ in the infinite norm and $$\mathcal{O}(n^{-m})$$ in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is $$\mathcal{O}(\log ^2 n)$$ and its spectral condition number is $$\mathcal{O}(1)$$ . Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Electrolyte-resistance change due to an insulating sphere in contact with a disk electrode

This work is aimed at improving the quantitative analysis of the electrolyte-resistance (ER) fluctuations generated by two-phase systems with dispersed gaseous, liquid, or solid insulating entities in a conductive electrolyte. The primary potential distribution around a disk electrode in contact with a small insulating sphere, which simulates a spherical particle, drop, or gas bubble sitting on a disk electrode used as a sensor, was calculated with a collocation method in order to derive the increment in ER caused by the sphere. For a sphere of size equal to one-tenth of the electrode size, the values of the ER increments were found to be very low and to depend on the sphere position: 0.3% close to the edge of the electrode and 0.05% at its centre. Despite the influence of variations in the electrolyte temperature and of the approximate horizontality of the electrode, these low values could be measured experimentally by scanning insulating spheres of 1 and 2 mm in diameter above or in contact with a stainless steel electrode of 10 mm in diameter, using the motorized translation stage of a scanning electrochemical microscope and a home-made electronic device measuring low-amplitude ER fluctuations.

A spectral solver for evolution problems with spatial [formula omitted]-topology

We introduce a single patch collocation method in order to compute solutions of initial value problems of partial differential equations whose spatial domains are 3-spheres. Besides the main ideas, we discuss issues related to our implementation and analyze numerical test applications. Our main interest lies in cosmological solutions of Einstein’s field equations. Motivated by this, we also elaborate on problems of our approach for general tensorial evolution equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy symmetry here.

A high order collocation method for the static and vibration analysis of composite plates using a first-order theory

A study of static deformations and free vibrations of shear flexible isotropic and laminated composite plates with a first-order shear deformation theory is presented. The analysis is based on collocation with a Deslaurier Dubuc interpolating basis to produce highly accurate results. Numerical results for isotropic and symmetric laminated composite plates are presented and discussed for various thickness-to-length ratios. The high order collocation method presented in this paper proved to be very accurate for this type of problems and the numerical efficiency is as good as other numerical schemes for first-order shear deformation theories, such as finite element solutions.

Convergence analysis for a second-order elliptic equation by a collocation method using scattered points

A collocation method using scattered points applied to a second-order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials. We show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method.

An optimal order collocation method for first kind boundary integral equations on polygons

This paper discusses the convergence of the collocation method using splines of any order $k$ for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ${\Bbb R}^2$ . Before discretization the equation is transformed to an equivalent equation over $[-\pi,\pi]$ using a nonlinear parametrization of the polygon which varies more slowly than arc--length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on $[-\pi,\pi]$ . This latter integral equation is shown to be well--posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non--standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order $k$ (degree $k-1$ ) converges with optimal order $O (h^k)$ . (The collocation points are the mid--points of subintervals when $k$ is odd and the break--points when $k$ is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples.

Application of a dual variational formulation to finite element reactor calculations

A number of improvements have been made to the nodal collocation method in order to obtain a high-order nodal technique capable of solving the neutron diffusion equation over full-core three-dimensional pressurized water reactors. First, the nodal collocation method is derived formally from a dual variational principle, using Gauss-Lobatto quadratures. Analytical integration and Gauss-Legendre quadratures are next applied to the same dual functional in order to obtain more accurate discretizations. An efficient ADI numerical technique with a supervectorization procedure was set up to solve the resulting matrix system. Validation results are given for the IAEA 2-D, Biblis and IAEA 3-D benchmarks and for a typical full-core 3-D representation of a pressurized water reactor at the beginning of the second cycle.

Sufficient conditions for a discrete maximum principle for high order collocation methods

The purpose of this work is to extend earlier results concerning a discrete maximum principle for collocation methods to higher order piecewise polynomial approximations. The techniques of this paper use a different set of basis elements for the piecewise polynomial space than that employed in the earlier work for the cubic and quartic cases.

An approximate analytical solution for the performance of reverse osmosis plants

Recent developments in membrane technology and appropriate construction material made reverse osmosis plant attractive for large desalting capacity. Reserve osmosis showed a growing demand specially in sea water desalting. It can be considered the optimum process in areas where sufficient electric power is available at low cost. Due to these reasons, mathematical modeling of reverse osmosis plants became an important task in the design procedure. In this paper, the partial differential equations representing the material and momentum balances inside a hollow fine fiber reverse osmosis model are discretized by the method of orthogonal collocation. The approximate analytical solution is obtained by applying the one point collocation method in the radial and axial direction. This leads to simple expressions for the recovery and product concentration. The obtained expressions are compared by the more exact results obtained by using higher order collocation method. The applicability of these results to the design and operation of reserve osmosis plants will be disucssed.

The performance of numerical methods for elliptic problems with mixed boundary conditions

AbstractWe consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5‐point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).

A Comparative Survey of Numerical Methods for the Linear Generalized Abel Integral Equation

AbstractA numerical comparison is made between a number of important representatives of the following classes of methods: (i) collocation, (ii) product integration and (iii) global methods. Special attention is paid to the performance of these methods for problems with a non‐smooth solution. It turned out that, when only relatively low accuracy is required, a good choice would be a second or third order collocation method of Branca. — A new collocation method which accounts for possible non‐smoothness of the solution near the origin, turned out to be advantageous when high accuracy is required, both for problems with a non‐smooth solution, and for problems with a smooth solution.

Quadratic Spline and Two-Point Boundary Value Problem

where f ( t , x, y) is defined and twice continuously differentiable in a region D of the (t, x} :y)-space intercepted by two hyperplanes £=0 and t=l. The analytical solution of (1.1) for arbitrary choices of the function / cannot be found in general. We usually resort to some numerical method for obtaining an approximate solution of the problem (1.1). The standard numerical methods for the numerical treatment of (1.1) consist of finite difference methods, shooting methods, RayleighRitz and Galerkin's procedure and collocation methods. A long list of references of all of these methods is given by Keller [4] in [2]. The subject of obtaining spline solutions for the initial as well as boundary value problems is briefly discussed in [1]. Since then many papers have appeared dealing with the continuous approximation of x(t) satisfying (1.1) via cubic and quintic splines mainly (see [3, 9]). The collocation methods using spline functions have been developed and analysed by Sakai (see [5], [6], [7]) again employing cubic and quintic splines at equi-distant knots. Recently Usmani and Sakai [11] have also used quadratic spline function for solving a two-point boundary value problem involving a fourth order differential equation. In this brief report, we propose a second order collocation method using quadratic spline employing 5-splines. Ill the sequel, it will be shown that our method is an 0(/i)-convergent procedure. In the last section some numerical evidence is included to show the practical applicability of our method by solving

On the Relative Efficiency of Higher Order Collocation Methods for Solving Two-Point Boundary Value Problems

Numerical results are presented which indicate that collocation methods using $C^1 $ piecewise polynomial functions with Gauss collocation points are more efficient than conventional second order finite difference methods for certain nonlinear two-point boundary value problems. For the problems studied, one obtains more accuracy per dollar of computer time with higher order collocation methods. Also for a given accuracy, higher order collocation methods require less computer storage than low order collocation methods or conventional second order finite difference methods.