A hybrid method for efficient solution of geometrically nonlinear structures

We propose a novel hybrid method for calculating accurate responses of geometrically nonlinear structures exhibiting complex snap-through and snap-back behaviours. The proposed method employs a hybrid evolutionary-algebraic method to obtain each point of the equilibrium path, where the equilibrium is formulated generally as a minimisation problem. Genetic algorithm and Nelder–Mead simplex methods are used together as the hybrid optimiser. To obtain any arbitrary point of the equilibrium path, we only need a series of sparse matrix to vector multiplications and do not require information about previous equilibrium states, the assembly of tangent stiffness matrix, the solution of a set of system of linear equations or factorisation processes. Both primary and secondary paths can be followed. In addition, utilising the tangent stiffness matrix, this method can effectively find both the limit and bifurcation points directly. The state of non-proportional loading can also be considered successfully. Additionally, we show how to generate the solution of a structure whose geometrical and mechanical properties vary slightly, starting from the original solution. Finally, to demonstrate the efficiency and capabilities of the present approach, three examples that are well known for their complex snap-through snap-back load–deflection curves are comprehensively studied, and the results obtained are compared with those reported in the literature.

► A novel hybrid solution method is proposed for geometrically nonlinear structures.
► Each point on the equilibrium path is found directly using an optimisation process.
► The method can effectively and directly find both the limit and bifurcation points.
► The state of non-proportional loading can be considered successfully.