Abstract
Solving systems of nonlinear equations is a difficult problem in numerical computation. Probably the best known and most widely used algorithm to solve a system of nonlinear equations is Newton-Raphson method. A significant shortcoming of this method becomes apparent when attempting to solve problems with limit points. Once a fixed load is defined in the first step, there is no way to modify the load vector should a limit point occur within the increment. To overcome this defect, displacement control methods for passing limit points can be used. In displacement control method, the load ratio in the first step of an increment is defined so that a particular key displacement component will change only by a prescribed amount. In this paper the load ratio is obtained using particle swarm optimization (PSO) algorithm so that the complex behavior of structures can be followed, automatically. Design variable is load ratio and its unbalanced force is also considered as objective function in optimization process. Numerical results are performed under geometrical nonlinear analysis, elastic post-buckling analysis and inelastic post-buckling analysis. The efficiency and accuracy of proposed method are demonstrated by solving these examples. Â
Highlights
The structural problems with geometrically nonlinear features can often be analyzed by solving a system of nonlinear equations to determine the path of nonlinear load-displacement
The results obtained by this paper show that the runtime of nonlinear analysis can be reduced by about 20–30% through the use of a hybrid method composed of Particle Swarm Optimization (PSO) and Newton-Raphson algorithms
These comparisons demonstrated the ability of the proposed method to return accurate solutions through a lower number of iterations and a shorter runtime
Summary
The structural problems with geometrically nonlinear features can often be analyzed by solving a system of nonlinear equations (algebraic or differential) to determine the path of nonlinear load-displacement. The literature on analysis of these problems is quite extensive and provides several approaches, most notably the incremental stiffness procedure [1], the perturbation method, the Newton-Raphson method and its modified variations, [1, 2], the initial value approach [3], and the self-correcting incremental procedure [1, 2], through all of which nonlinear equilibrium equations could be solved in an efficient manner
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