Constrained Optimization Shooting Method for Predicting the Periodic Solutions of Nonlinear System

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Haitao Liao
Keyword(s):  
Nonlinear System ◽  
Shooting Method ◽  
Continuation Method ◽  
Inequality Constraints ◽  
Original Method ◽  
Maximization Problem ◽  
Numerical Examples ◽  
Equality And Inequality Constraints ◽  
Parameter Continuation ◽  
Nonlinear Equality

An original method for calculating the maximum vibration amplitude of the periodic solution of a nonlinear system is presented. The problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem. The shooting method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness and ability of the proposed approach are illustrated through two numerical examples. Numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated.

A Weighted Flexible Tolerance Algorithm for Constrained Minimization

1975 ◽  
Vol 97 (4) ◽  
pp. 1291-1294 ◽  
Author(s):  
M. A. Townsend ◽  
F. Y. Lam
Keyword(s):  
Objective Function ◽  
Search Algorithm ◽  
Inequality Constraints ◽  
Constrained Minimization ◽  
Rapid Convergence ◽  
Flexible Polyhedron ◽  
Equality And Inequality Constraints ◽  
Direct Search Algorithm ◽  
Tolerance Method ◽  
Nonlinear Equality

A direct search algorithm based upon the flexible tolerance method developed by Paviani and Himmelblau [1] for minimization of a functional subject to nonlinear equality and inequality constraints is proposed and evaluated. The modification incorporates a weighting on the direction of search in which the flexible polyhedron formed orients itself toward the gradient of the objective function (and clings to all active constraints). This is accomplished by weighting the centroid, in the sense of minimization, of the polyhedron with respect to the objective function. A number of problems have been solved satisfactorily by this algorithm, generally with more rapid convergence.

scholarly journals Uncertainty Quantification and Bifurcation Analysis of an Airfoil with Multiple Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Haitao Liao
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance Method ◽  
Inequality Constraints ◽  
Maximization Problem ◽  
Limit Cycle Oscillation ◽  
Structural Parameter ◽  
Limit Cycle Oscillations ◽  
Aeroelastic System ◽  
Equality And Inequality Constraints ◽  
The Stability

In order to calculate the limit cycle oscillations and bifurcations of nonlinear aeroelastic system, the problem of finding periodic solutions with maximum vibration amplitude is transformed into a nonlinear optimization problem. An algebraic system of equations obtained by the harmonic balance method and the stability condition derived from the Floquet theory are used to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the proposed approach is validated, and the effects of structural parameter uncertainty on the limit cycle oscillations and bifurcations of an airfoil with multiple nonlinearities are studied. Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.

scholarly journals Efficiency and duality for multiobjective fractional variational problems with (ρ,b) - quasiinvexity

2009 ◽  
Vol 19 (1) ◽  
pp. 85-99 ◽  
Author(s):  
Ştefan Mititelu ◽  
I.M. Stancu-Minasian
Keyword(s):  
Necessary Conditions ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Efficient Solutions ◽  
Parametric Approach ◽  
Duality Theorems ◽  
Normal Efficiency ◽  
Fractional Variational Problems ◽  
Equality And Inequality Constraints ◽  
Nonlinear Equality

The necessary conditions for (normal) efficient solutions to a class of multi-objective fractional variational problems (MFP) with nonlinear equality and inequality constraints are established using a parametric approach to relate efficient solutions of a fractional problem and a non-fractional problem. Based on these normal efficiency criteria a Mond-Weir type dual is formulated and appropriate duality theorems are proved assuming (?,b) - quasi-invexity of the functions involved.

scholarly journals On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions

2021 ◽  
Vol 19 (1) ◽  
pp. 760-772
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Badrah Alghamdi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential

Abstract We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

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