• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.26 no.7
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• pp.795-805
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• 2016

Nonlinear normal modes(NNMs) is a branch of periodic solution of nonlinear dynamic systems. Determination of stable periodic solution is very important in many engineering applications since the stable periodic solution can be an attractor of such nonlinear systems. Periodic solutions of nonlinear system are usually calculated by perturbation methods and numerical methods. In this study, numerical method is used in order to calculate the NNMs. Iteration of the solution is presented by multiple shooting method and continuation of solution is presented by pseudo-arclength continuation method. The stability of the NNMs is analyzed using Floquet multipliers, and bifurcation points are calculated using indirect method. Proposed analyses are applied to two nonlinear numerical models. In the first numerical model nonlinear spring-mass system is analyzed. In the second numerical model Jeffcott rotor system which has unstable equilibria is analyzed. Numerical simulation results show that the multiple shooting method can be applied to self excited system as well as the typical nonlinear system with stable equilibria.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.2
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• pp.85-97
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• 2002

In this paper, finite volume element methods for nonlinear parabolic problems are proposed and analyzed. Optimal order error estimates in $W^{1,p}$ and $L_p$ are derived for $2{\leq}p{\leq}{\infty}$. In addition, superconvergence for the error between the approximation solution and the generalized elliptic projection of the exact solution (or and the finite element solution) is also obtained.

• Structural Engineering and Mechanics
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• v.48 no.6
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• pp.879-914
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• 2013

In part I of the article, formulation and characteristics of the several well-known structural geometrical nonlinear solution techniques were studied. In the present paper, the efficiencies and capabilities of residual load minimization, normal plane, updated normal plane, cylindrical arc length, work control, residual displacement minimization, generalized displacement control and modified normal flow will be evaluated. To achieve this goal, a comprehensive comparison of these solution methods will be performed. Due to limit page of the article, only the findings of 17 numerical problems, including 2-D and 3-D trusses, 2-D and 3-D frames, and shells, will be presented. Performance of the solution strategies will be considered by doing more than 12500 nonlinear analyses, and conclusions will be drawn based on the outcomes. Most of the mentioned structures have complex nonlinear behavior, including load limit and snap-back points. In this investigation, criteria like number of diverged and complete analyses, the ability of passing load limit and snap-back points, the total number of steps and analysis iterations, the analysis running time and divergence points will be examined. Numerical properties of each problem, like, maximum allowed iteration, divergence tolerance, maximum and minimum size of the load factor, load increment changes and the target point will be selected in such a way that comparison result to be highly reliable. Following this, capabilities and deficiencies of each solution technique will be surveyed in comparison with the other ones, and superior solution schemes will be introduced.

• Journal of applied mathematics & informatics
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• v.6 no.1
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• pp.175-184
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• 1999

Along with the computation and analysis for nonlinear system being more and more involved in the fields such as automation control electronic technique and electrical power system the nonlin-ear theory has become quite a attractive field for academic research. In this paper we derives the solutions for state equation of nonlinear system by using the inverse operator expression of the so-lutions is obtained. An actual computation example is given giving a comparison between IOM and Runge-kutta method. It has been proved by our investigation that IOM has some distinct advantages over usual approximation methods in that it is computationally con-venient rapidly convergent provides accurate solutions not requiring perturbation linearization or the massive computation inherent in discrietization methods such as finite differences. So the IOM pro-vides an effective method for the solution of nonlinear system is of potential application valuable in nonlinear computation.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.6 no.3
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• pp.11-20
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• 1986

The purpose of this study is to devise an efficient and economic solution algorithm for the nonlinear finite element equations. First, procedures and characteristics of the solution methods of ordinary nonlinear equations are critically reviewed and discussed. Based on the discussion, some promising nonlinear finite element analysis procedures are presented as an algorithmic form. Finally, a conceptually combined algorithm for a solution of nonlinear finite element equations is proposed and analyzed, in which the computational effort is minimized and numerical difficulties can be avoided.

• Structural Engineering and Mechanics
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• v.83 no.2
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• pp.135-151
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• 2022

The generalized displacement method is a nonlinear solution scheme that follows the equilibrium path of the structure based on the development of the generalized displacement. This method traces the path uniformly with a constant amount of generalized displacement. In this article, we first develop higher-order generalized displacement methods based on multi-point techniques. According to the concept of generalized stiffness, a relation is proposed to adjust the generalized displacement during the path-following. This formulation provides the possibility to change the amount of generalized displacement along the path due to changes in generalized stiffness. We, then, introduce higher-order algorithms of variable generalized displacement method using multi-point methods. Finally, we demonstrate with numerical examples that the presented algorithms, including multi-point generalized displacement methods and multi-point variable generalized displacement methods, are capable of following the equilibrium path. A comparison with the arc length method, generalized displacement method, and multi-point arc-length methods illustrates that the adjustment of generalized displacement significantly reduces the number of steps during the path-following. We also demonstrate that the application of multi-point methods reduces the number of iterations.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.329-336
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• 2006

In the category of the group theoretic methods using invertible discrete group transformation, we give a useful relation between Emden-Fowler equations and nonlinear heat equation. In this paper, by means of appropriate transformations of discrete group analysis, the nonlinear hate equation transformed into the class of the Emden-Fowler equations. This approach shows that, under the group action, the solution of reference equation can be transformed into the solution of the transformed equation.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.745-756
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• 2022

We study a nonlinear wave equation on finite connected weighted graphs. Using Rothe's and energy methods, we prove the existence and uniqueness of solution under certain assumption. For linear wave equation on graphs, Lin and Xie [10] obtained the existence and uniqueness of solution. The main novelty of this paper is that the wave equation we considered has the nonlinear damping term |u_t|^p-1·u_t (p > 1).

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.669-676
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• 2012

In this note we extend one well-known iteration formula to derive new third-order methods for solving a nonlinear equation. The comparison with other methods is given.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10a
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• pp.314-317
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• 1993

In this paper, the IPE(Iterative Parameter Estimation) methods for the nonlinear measurements are proposed. The IPE methods convert the problems of the parameter estimation for the nonlinear measurements to that of the solution of the nonlinear equations approximately and use several iterative numerical solutions, such as fixed points theory, Newton's methods, quasi-Newton's methods and steepest descent techniques. the IPE methods for the nonlinear measurements-in the case of the error estimation for the inertial navigation systems are simulated, and it is found that the estimation errors for the nonlinear measurements decrease rapidly and converge to almost that of the linear LSE(Least Squares Estimation) when the IPE methods are applied.