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1.
    
In this paper, an exponentially fitted method is used to numerically solve the one‐dimensional Burgers' equation. The performance of the method is tested on the model involving moderately large Reynolds numbers. The obtained numerical results show that the method is efficient, stable and reliable for solving Burgers' equation accurately even involving high Reynolds numbers for which the exact solution fails. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

2.
    
In this paper we propose the numerical solution of a steady‐state reaction‐diffusion problem by means of application of a non‐local Lyapunov–Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non‐linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non‐linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple non‐linear one‐dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

3.
    
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.  相似文献   

4.
    
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.  相似文献   

5.
    
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.  相似文献   

6.
    
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.  相似文献   

7.
    
This paper is concerned with the minimization of functionals of the form ∫Γ(b) f( h ,T( b, h )) dΓ( b ) where variation of the vector b modifies the shape of the domain Ω on which the potential problem, ?2T=0, is defined. The vector h is dependent on non‐linear boundary conditions that are defined on the boundary Γ. The method proposed is founded on the material derivative adjoint variable method traditionally used for shape optimization. Attention is restricted to problems where the shape of Γ is described by a boundary element mesh, where nodal co‐ordinates are used in the definition of b . Propositions are presented to show how design sensitivities for the modified functional ∫Γ(b) f( h ,T ( b, h )) dΓ( b ) +∫Ω(b) λ( b, h ) ?2T( b, h ) dΩ( b ) can be derived more readily with knowledge of the form of the adjoint function λ determined via non‐shape variations. The methods developed in the paper are applied to a problem in pressure die casting, where the objective is the determination of cooling channel shapes for optimum cooling. The results of the method are shown to be highly convergent. Copyright © 2002 John Wiley & Sons, Ltd.  相似文献   

8.
    
In this paper we propose a change in the representation of the discrete motion equations in structural non‐linear dynamics to obtain an improvement in the stability of time numerical integrations. In particular, natural local state variables are indicated for a finite element approach to beam problems. The results, relative to Newmark approximations for the variations in the displacement and velocity vectors, show a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration steps. The proposed method, further, preserves energy as well as the linear and angular momentum of the dynamical system. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

9.
    
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.  相似文献   

10.
    
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.  相似文献   

11.
    
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.  相似文献   

12.
    
We present the development of a two‐dimensional Mixed‐Hybrid Finite Element (MHFE) model for the solution of the non‐linear equation of variably saturated flow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest‐order Raviart–Thomas (RT0) elements and is ‘exactly’ mass conserving. Hybridization is used to overcome the ill‐conditioning of the mixed system. The scheme is globally first‐order in space. Nevertheless, numerical results employing non‐uniform meshes show second‐order accuracy of the pressure head and normal fluxes on specific grid points. The non‐linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE‐Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques. Copyright © 1999 John Wiley & Sons, Ltd.  相似文献   

13.
    
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.  相似文献   

14.
    
Very often, when dealing with computational methods in engineering analysis, the final state depends so sensitively on the system's precise initial conditions that the behaviour becomes unpredictable and cannot be distinguished from a random process. This outcome is rooted in an intricate phenomenon labelled ‘chaos’, which is a synonym for unpredictable events in nature. In contrast, chaos is a deterministic feature that can be utilized for problems of finding global solutions in both non‐linear systems of equations as well as optimization. The focus of this paper is an attempt to utilize computational instabilities in solving systems of non‐linear equations and optimization theory that resulted in development of a new method, chaotic descent. The method is based on descending to global minima via regions that are the source of computational chaos. Also, one very important conjecture is presented that in the future might lead the way towards direct solving of the systems of simultaneous non‐linear equations for all the solutions. Copyright © 2000 John Wiley & Sons, Ltd.  相似文献   

15.
    
This paper presents a new family of time‐stepping algorithms for the integration of the dynamics of non‐linear shells. We consider the geometrically exact shell theory involving an inextensible director field (the so‐called five‐parameter shell model). The main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid. In this context, we develop time‐stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system, and allow the introduction of a controllable non‐negative energy dissipation to handle the high numerical stiffness characteristic of these problems. A series of algorithmic parameters for the different components of the deformation of the shell (i.e. membrane, bending and transverse shear) fully control this numerical dissipation, recovering existing energy‐momentum schemes as a particular choice of these algorithmic parameters. We present rigorous proofs of the numerical properties of the resulting algorithms in the full non‐linear range. Furthermore, it is argued that the numerical dissipation is introduced in the high‐frequency range by considering the proposed algorithm in the context of a linear problem. The finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation/dissipation properties. We present several representative numerical simulations illustrating the performance of the newly proposed methods. The robustness gained over existing methods in these stiff problems is confirmed in particular. Copyright © 2002 John Wiley & Sons, Ltd.  相似文献   

16.
    
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.  相似文献   

17.
    
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.  相似文献   

18.
    
The accuracy problem of the semi‐analytical method for shape design sensitivity analysis has been reported for linear and non‐linear structures. The source of error is the numerical differentiation of the element internal force vector, which is inherent to the semi‐analytical approach. Such errors occur for structures whose displacement field is characterized by large rigid body rotations of individual elements. This paper presents a method for the improvement of semi‐analytical sensitivities. The method is based on the element free body equilibrium conditions, and on the exact differentiation of the rigid body modes. The method is efficient, simple to code, and can be applied to linear and non‐linear structures. The numerical examples show that this approach eliminates the abnormal errors that occur in the conventional semi‐analytical method. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

19.
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20.
    
A sinh transformation has recently been proposed to improve the numerical accuracy of evaluating nearly singular integrals using Gauss–Legendre quadrature. It was shown that the transformation could improve the accuracy of evaluating such integrals, which arise in the boundary element method, by several orders of magnitude. Here, this transformation is extended in an iterative fashion to allow the accurate evaluation of similar types of integrals that have more spiked integrands. Results show that one iteration of this sinh transformation is preferred for nearly weakly singular integrals, whereas two iterations lead to several orders of magnitude improvement in the evaluation of nearly strongly singular integrals. The same observation applies when considering integrals of derivatives of the two‐dimensional boundary element kernel. However, for these integrals, more iterations are required as the distance from the source point to the boundary element decreases. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

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