A homotopy analysis method for the nonlinear partial differential equations arising in engineering

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.     

The Global Nonlinear Galerkin Method for the Analysis of Elastic Large Deflections of Plates under Combined Loads: A Scalar Homotopy Method for the Direct Solution of Nonlinear Algebraic Equations

In this paper, the global nonlinear Galerkin method is used to perform an accurate and efficient analysis of the large deflection behavior of a simply-supported rectangular plate under combined loads. Through applying the Galerkin method to the governing nonlinear partial differential equations (PDEs) of the plate, we derive a system of coupled third order nonlinear algebraic equations (NAEs). However, the resultant system of NAEs is thought to be hard to tackle because one has to find the one physical solution from among the possible multiple solutions. Therefore, a suitable initial guess is required to lead to the real solution for given load conditions. The feature of this paper is that we apply the global nonlinear Galerkin method to the governing PDEs and solve the resultant NAEs directly in each load step. To keep track of the physical solution, the initial guess for the current load step is provided by taking the solution of the NAEs for the last step as the initial guess. Besides, the size of the NAEs grows dramatically larger, with the increase of the number of terms of the trial functions, which will cost much more computational efforts. An exponentially convergent scalar homotopy algorithm (ECSHA) is introduced to solve the large set of NAEs. The approach in the present paper is more direct and simpler than either the incremental global Galerkin method, or the incremental local Galerkin method (finite element method) based on a symmetric incremental weak-form; both of which methods lead to the inversion of tangent stiffness matrices and Newton-Raphson iterations in each load step. The present method of exponentially convergent scalar homotopy of directly solving the NAEs is much better than the quadratically convergent Newton-Raphson method. Several numerical examples are provided to validate the feasibility and efficiency of the proposed scheme.     

基于同伦延拓的冗余机械臂逆运动学优化算法研究

目的 针对偏置冗余机械臂的逆运动学,采用传统数值法存在依赖初始值、奇异位姿收敛性差等问题,提出一种改进数值法。方法 首先将非线性方程组转化为同伦方程组,引入同伦延拓算法能够有效避免依赖初始值的问题,同时能够获取逆运动学解空间。然后考虑奇异位姿,将同伦方程组转化为最小二乘问题,采用Levenberg Marquardt算法对同伦方程组进行路径追踪,以获取逆运动学解空间。最后将关节极限避免问题映射为解空间优化问题,引入二进制改进粒子群优化算法,获得最优逆运动学解。结果 实验结果表明,相较于传统数值法,文中所提数值法针对逆运动学求解具有更高的收敛率、更快的收敛速度,同时二进制改进粒子群算法能够有效避免关节极限问题。结论 采用文中所提数值法求解逆运动学的精度较高,能够满足实时性要求,对于机械臂用于包装作业具有一定的理论意义和工程应用价值。     

On the purely analytic computation of laminar boundary layer flow over a rotating cone

The fundamental purpose of the present research is to obtain analytical expressions for the solution of the steady laminar flow of an incompressible viscous Newtonian fluid over a rotating cone. Using a proper similarity transformation akin to the classical one of Von Karman the nonlinear equations of motion are reduced to a boundary value problem whose solution is then derived in terms of a series of exponentially-decaying functions for the full range of cone half-angle ? characterizing the conical flow structure. The exact numerical method is found to improve as the cone half-angle is decreased. The effects of the cone half-angle on the physically significant relevant parameters, such as the wall shears, the torque and the vertical suction are clarified. Purely explicit analytical expressions for the solution of governing equations to support the numerically evaluated solutions are also obtained via the homotopy analysis method.     

Extended homotopy analysis method for multi-degree-of-freedom non-autonomous nonlinear dynamical systems and its application

In normal circumstances, numerous practical engineering problems are multi-degree-of-freedom (MDOF) nonlinear non-autonomous dynamical systems. Generally, exact solutions for MDOF dynamical systems are hardly obtained; thus, the development of analytical approximations becomes a robust and appealing avenue for an analysis of these systems. The homotopy analysis method (HAM) is one of the analytical methods, which can overcome the foregoing barriers of conventional asymptotic techniques. It has been widely used for solving various nonlinear problems in physical science and engineering. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for MDOF weakly damped non-autonomous dynamical systems. In terms of its flexibility and applicability, the EHAM is also applied to derive the approximate solutions of parametrically and externally excited thin plate systems. Besides, comparisons are performed between the results obtained by the EHAM and the numerical integration (i.e. Runge–Kutta) method. The present findings show that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions.     

双层梁在均匀升温场内的几何非线性问题

基于精确的几何非线性理论,建立了轴线可伸长双层梁在温度载荷作用下的非线性弯曲控制方程。其中包含了由于材料在横向非均匀分布而导致的拉-弯耦合项。应用打靶法数值求解相应的非线性边值问题,得到了均匀加热下两端不可移简支双层梁的热弯曲数值解。作为算例,给出了由铜和钢组成的双金属梁的平衡构形和平衡路径,分析和讨论了几何和物理参数对梁变形的影响。     

3D symplectic expansion for piezoelectric media

The symplectic series expansion method is extended to three‐dimensional problem for transversely isotropic piezoelectric media. The governing equations are first derived in Hamiltonian form, and symplectic eigensolutions are directly obtained through analytical method. All solutions of the problem are reduced to finding eigenvalues and eigensolutions. The classical St Venant solutions are described by zero‐eigensolutions, and the localized solutions are depicted by non‐zero‐eigensolutions. Symplectic relationships of the ortho‐normalization are used, end conditions are rewritten by eigensolutions, and a numerical scheme is formed analytically. Some numerical examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way. In it, we introduce a method called residual correction procedure, to correct some previous approximate solutions for such systems. We study the error analysis of our given method. We first introduce a new result to approximate the absolute solution by using the residual correction procedure. Second, we introduce a new result to get appropriate bound for the absolute error. The collocation method is used and the collocation points can be found by applying Chebyshev roots. Both techniques are explained briefly with illustrative examples to demonstrate the applicability, efficiency and accuracy of the techniques. By using a small number of Bernstein polynomials and correction procedure we achieve some significant results. We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method, continuous genetic algorithm, rational homotopy perturbation method and adomian decomposition method.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

Active stiffness control of wind‐rain‐induced vibration of prototype stay cable

The cables in a cable‐stayed bridge usually possess low inherent damping and are prone to wind‐induced, traffic‐induced, and wind‐rain‐induced vibrations. This paper establishes an active control algorithm using the stiffness control method to suppress wind‐rain‐induced vibration of prototype stay cables. By neglecting the axial inertia force and the modal coupling, the governing equations of motion of wind‐rain‐induced vibration control of prototype stay cables with active stiffness control algorithm are first derived. The fourth‐order Runge–Kutta method is then introduced to find the numerical solutions to the problem. Extensive parameter studies have been carried out for investigating the features of the control method as a design guideline. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability analysis of arbitrary shape landslope by variational calculus and finite difference method

Abstract

In this paper, a method of numerical analysis is presented, which is suitable for the stability analysis of arbitrary shape landslope, and can be used for the stability analysis of both general artificial landslope and arbitrary natural landslope. The mathematical model of stability analysis is based on the theory of the calculus of variations and using Janbu's simplified model. The governing equations are a system of integro‐differential equations which consist of Euler's equations together with the transversality, continuity and boundary conditions. Applying the concept of finite difference method, we set up the numerical procedure for this system. The numerical procedure presented in this paper is powerful for determining the critical sliding surface of arbitrary shape landslope without the necessity of guessing its shape. Furthermore, it gives the actual safety factor of the landslope and consequently obtains a more complete knowledge of the safety of arbitrary shape landslope. For comparison, the proposed numerical method is applied to analyze two problems which have analytic solutions. The numerical results have shown highly accurate. Some examples including the double rectilinear and arbitrary natural landslopes are analyzed inorder to illustrate the applicability of this numerical method.     

On the contact mechanics of a rolling cylinder on a graded coating. Part 1: Analytical formulation

In this paper, the fully coupled rolling contact problem of a graded coating/substrate system under the action of a rigid cylinder is investigated. Using the singular integral equation approach, the governing equations of the rolling contact problem are constructed for all possible stick/slip regimes. Applying the Gauss–Chebyshev numerical integration method, the governing equations are converted to systems of algebraic equations. A new numerical algorithm is proposed to solve these systems of equations. Both the coupled and the uncoupled solutions to the problem are found through an implemented iterative procedure. In Part I of this paper, the analytical formulation of the rolling contact problem and the discretization of the governing equations are introduced for all assumed stick/slip regimes. A detailed discussion of the proposed numerical algorithm, the iteration procedure and the numerical results, obtained using the analytical formulation, are given in Part II.     

New electromagnetic lift control method for magnetic levitation systems and magnetic bearings

The classical approach to gap control in active magnetic bearings-including those in magnetic levitation (maglev) systems-is proportional-integral-derivative (pid) based current correction. This paper explores a new method that simplifies control electronics, based on repeatedly solving the governing system equations in approximations that are valid for the next 20 to 40 ms. The method simplifies the magnetic forces by using a Taylor approximation, one that can be evaluated rapidly by using multivariate splines. The simplified equations of motion are solved by the method of Frobenius. These simplified solutions are inverted to predict the voltage necessary to achieve a desired gap change in a specified time increment. Variations from this target position allow for an update on inertia and mass of the levitated object.     

Conservation of energy for schemes applied to the propagation of shallow‐water inertia‐gravity waves in regions with varying depth

The linear equations governing the propagation of inertia‐gravity waves in geophysical fluid flows are discretized on the Arakawa C‐grid using centered differences in space. In contrast to the constant depth case it is demonstrated that varying depth may give rise to increasing energy (and loss of stability) using the natural approximations for the Coriolis terms found in many well‐known codes. This is true no matter which numerical method is used to propagate the equations. By a simple trick based on a modified weighting that ensures that the propagation matrices for the spatially discretized equations become similar to skew‐symmetric matrices, this problem is removed and the energy is conserved in regions with varying depth too. We give a number of examples both of model problems and large‐scale problems in order to illustrate this behaviour. In real applications diffusion, explicit through frictional terms or implicit through numerical diffusion, is introduced both for physical reasons, but often also in order to stabilize the numerical experiments. The growing modes associated with varying depth, the C‐grid and equal weighting may force us to enhance the diffusion more than we would like from physical considerations. The modified weighting offers a simple solution to this problem. Copyright © 2000 John Wiley & Sons, Ltd.     

New formulae for switch ed‐capacitor band pass filters

Abstract

The governing equations for switched‐capacitor band pass filters are greatly simplified into a new set of much simpler equations, with no approximation involved. For sharp band pass filters with narrow bandwidths, we expand the new set, cancel out many leading terms, and get a set of new formulae. Based on these formulae, we obtain and tabulate numerical design data for sharp 4th to 8th order Butterworth band pass filters, including values for the required resistors. The application of switched‐capacitor filters MF5 and MF10 is then greatly expanded to cover sharp and very sharp band pass filters.     

A simplified two-fluid theory for helium(II)

We construct the equations for the incompressible two-fluid model of He(II). These equations provide a set of partial differential equations governing the thermomechanical behaviour of liquid helium below the λ-point. This set of equations is similar but slightly simpler than the equations originally proposed by Landau [1]. Following the standard approach of continuum mixture theory, we derive thermodynamic restrictions on the incompressible two-fluid model. Finally, we establish conditions which ensure that classical solutions of certain initial-boundary value problems for the simplified theory depend continuously on the initial data.     

Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero‐velocities in solid material. The resulting parameter optimization problem is solved by a non‐linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design‐dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non‐physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady‐state counterparts. Copyright © 2011 John Wiley & Sons, Ltd.     

A two-dimensional problem for a fibre-reinforced anisotropic thermoelastic half-space with energy dissipation

The theory of thermoelasticity with energy dissipation is employed to study plane waves in a fibre-reinforced anisotropic thermoelastic half-space. We apply a thermal shock on the surface of the half-space which is taken to be traction free. The problem is solved numerically using a finite element method. Moreover, the numerical solutions of the non-dimensional governing partial differential equations of the problem are shown graphically. Comparisons are made with the results predicted by Green–Naghdi theory of the two types (GNII without energy dissipation) and (GNIII with energy dissipation). We found that the reinforcement has great effect on the distribution of field quantities. Results carried out in this paper can be used to design various fibre-reinforced anisotropic thermoelastic elements under thermal load to meet special engineering requirements.