Numerical solution of non‐linear Klein–Gordon equations by variational iteration method

In this paper, numerical solution of non‐linear Klein–Gordon equations with power law non‐linearities are obtained by the new application of He's variational iteration method. Numerical illustrations that include non‐linear Klein–Gordon equations and non‐linear partial differential equations are investigated to show the pertinent features of the technique. Copyright © 2006 John Wiley & Sons, Ltd.     

Solving systems of non‐linear equations with both free and positive variables

A numerical method is presented for solving systems of non‐linear equations that contain some variables that are strictly positive and others that have no restriction on sign. Naturally positive variables arise frequently when modelling the behaviour of engineering systems, such as physical dimension, concentration of a chemical species, duration of an event, etc. When modelling systems of this type, it is also common to introduce additional variables that are not restricted in sign, such as stresses, displacements, velocities, accelerations, etc. Many numerical methods may experience performance difficulties due to the existence of spurious solutions which have negative components for one or more of the positive variables. Recently, the monomial method has been developed as an effective tool for systems with variables that are all strictly positive. This paper presents a hybrid method, combining the monomial method and Newton's method, for systems containing both types of variables. It is demonstrated that this hybrid method can be more effective in solving systems of equations with both positive and free variables than either method alone. Basins of attraction constructions are presented as a demonstration of the effectiveness of the hybrid method as applied to the design of a civil engineering frame structure. Copyright © 1999 John Wiley & Sons, Ltd.     

Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical method for non‐linear wave and diffusion equations by the variational iteration method

This paper applies He's variational iteration to the wave equations in an infinite one‐dimensional medium and some non‐linear diffusion equations. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

Relaxation iterative algorithms for solving cathodic protection systems with non‐linear polarization curves

This paper discusses the calculation of potential distribution of impressed cathodic protection (CP) models with non‐linear polarization curves. We propose a relaxation iterative algorithm for the non‐linear problem and prove both theoretically and numerically that this iterative sequence is convergent for any physical polarization curves. This feature is of significant importance in developing a computer code for the design of CP systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

The consistency and stability of a Thomas–Gladwell family of multistage time‐stepping schemes for the solution of first‐order non‐linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second‐order governing equations. Second‐order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non‐linear coefficients and is exploited to develop a new non‐iterative modification of the Thomas–Gladwell method that is second‐order accurate and unconditionally stable. A case study from applied hydrogeology using the non‐linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non‐iterative formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method

In this paper, we consider linear and non‐linear space–time fractional reaction–diffusion equations (STFRDE) on a finite domain. The equations are obtained from standard reaction–diffusion equations by replacing a second‐order space derivative by a fractional derivative of order β∈(1, 2], and a first‐order time derivative by a fractional derivative of order α∈(0, 1]. We use the Adomian decomposition method to construct explicit solutions of the linear and non‐linear STFRDE. Finally, some examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

A Generalised Conjugate Residual method for the solution of non‐symmetric systems of equations with multiple right‐hand sides

This paper presents an iterative algorithm for solving non‐symmetric systems of equations with multiple right‐hand sides. The algorithm is an extension of the Generalised Conjugate Residual method (GCR) and combines the advantages of a direct solver with those of an iterative solver: it does not have to restart from scratch for every right‐hand side, it tends to require less memory than a direct solver, and it can be implemented efficiently on a parallel computer. We will show that the extended GCR algorithm can be competitive with a direct solver when running on a single processor. We will also show that the algorithm performs well on a Cray T3E parallel computer. Copyright © 1999 John Wiley & Sons, Ltd.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient second‐order characteristic finite element method for non‐linear aerosol dynamic equations

In the paper we consider the non‐linear aerosol dynamic equation on time and particle size, which contains the advection process of condensation growth and the process of non‐linear coagulation. We develop an efficient second‐order characteristic finite element method for solving the problem. A high accurate characteristic method is proposed to treat the condensation advection while a second‐order extrapolation along the characteristics is proposed to approximate the non‐linear coagulation. The method has second‐order accuracy in time and the optimal‐order accuracy of finite element spaces in particle size, which improves the first‐order accuracy in time of the classical characteristic method. Numerical experiments show the efficient performance of our method for problems of log‐normal distribution aerosols in both the Euler coordinates and the logarithmic coordinates. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical procedure to solve non‐linear kinematic problems in spatial mechanisms

This paper presents a numerical method to solve the forward position problem in spatial mechanisms. The method may be incorporated in a software for the kinematic analysis of mechanisms, where the procedure is systematic and can be easily implemented, achieving a high degree of automation in simulation. The procedure presents high computational efficiency, enabling its incorporation in the control loop to solve the forward position problem in the case of a velocity control scheme. Also, in this paper preliminary results on the convergence of the proposed procedure are shown, and efficiency results of the method applied to representative spatial mechanisms are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive Kronrod–Patterson integration of non‐linear finite element matrices

Efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields. The highly non‐linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result. It is shown that these strong variations conflict with the common preference for low‐order numerical integration in finite element simulations of unsaturated moisture flow: inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization. In response, this article develops adaptive integration, based on nested Kronrod–Patterson–Gauss integration schemes: basically, the integration order is adapted to the locally observed grade of non‐linearity. Adaptive integration is developed based on a standard infiltration problem, and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained, thus significantly increasing computational efficiency. The multi‐dimensional applicability is exemplified with two‐dimensional wetting and drying applications. While developed for finite element unsaturated moisture transfer simulation, adaptive integration is similarly applicable for other non‐linear problems and other discretization methods, and whereas perhaps outperformed by mesh‐adaptive techniques, adaptive integration requires much less implementation and computation. Both techniques can moreover be easily combined. Copyright © 2009 John Wiley & Sons, Ltd.     

A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown‐field derivatives and have particular appeal in problems involving high‐order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197 :2901–2929) to develop a formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one‐field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non‐linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple‐stress from the displacement field at the mid‐surface of the shell, the non‐linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple‐stress at the edges of the shells, the extension to non‐linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.     

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

This paper presents a study of the performance of the non‐linear co‐ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non‐linear polynomial transformations is presented for two‐dimensional problems. Effectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more efficiently handled with transformations valid for end‐point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non‐linear transformation has been proposed. This composite approach is found to be more accurate, efficient and robust than the singularity subtraction method and the non‐linear transformation methods. Copyright © 2001 John Wiley & Sons, Ltd.