Keywords: stability-constrained optimization, shallow trusses, higher order pathfollowing method, hybrid heuristic method.

In this paper, a higher order path-following method is presented to tackle the structural stability constraints within truss optimisation. The general truss optimization problem is formulated as a discrete or continuous minimal weight design subjected to several constraints which require the computation of the structural response.

The proposed path-following method [1] is based on the perturbation technique of the stability theory [2] and a non-linear modification of the classical linear homotopy method [3]. The nonlinear function of the total potential energy for conservative systems can be expressed in terms of nodal displacements and the load parameter. The equilibrium equations are given from the principle of the stationary value of the total potential energy. The stability investigation is based on the eigenvalue computation of the Hessian matrix. In each step of the path-following process we get information about the displacement, stresses, local, and global stability of the structure. With the help of the higher-order predictor-corrector algorithm, we are able to compute an arbitrary load deflection path and detect the different type of stability points. During the optimization process, a truss is characterized by its maximal load intensity factor along the equilibrium path.

Within the predictor step, an implicit ODE problem and in the corrector phase a non-linear equation system has to be solved. The first-order derivative is obtained from the null-space of the augmented Jacobian matrix. The higher order derivatives need the Moor-Penrose pseudo-inverse of the augmented Jacobian matrix. The equilibrium path computation terminates if the procedure reaches the applied load level, a singular point or any other constraints violating point on the equilibrium path.

In order to demonstrate the viability and the efficiency of the proposed approach computational results are presented using the ANGEL hybrid metaheuristic as the optimization method.

1

A. Csébfalvi, "A non-linear path-following method for computing the equilibrium curve of structures", Annals of Operation Research, 81, 15-23, 1998. doi:10.1023/A:1018944804979

2

J.M.T. Thompson, G.W. Hunt, "Elastic Instability Phenomena", Wiley, New York, 1984. doi:10.1115/1.3169017

3

E.L. Allgower, "A survey of homotopy methods for smooth mappings", in E.L. Allgower et al., (Editors), "Lecture Notes in Mathematics-Numerical Solution of Nonlinear Equations", Springer-Verlag, Berlin, 2-29, 1981. doi:10.1007/BFb0090675

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