Structural optimization by spherical interpolation of objective function and constraints

A new method for structural optimization is presented for successive approximation of the objective function and constraints in conjunction with Lagrange multipliers approach. The focus is on presenting the methodology with simple examples. The basis of the iterative algorithm is that after each iteration, it brings the approximate location of the estimated minimum closer to the exact location, gradually. In other words, instead of the linear or parabolic term used in Taylor expansion, which works based on a short step length, an arch is used that has a constant curvature but a longer step length. Using this approximation, the equations of optimization involve the Lagrange multipliers as the only unknown variables. The equations which depend on the design variables are decoupled linearly as these variables are directly obtained. One mathematical example is solved to explain in details how the method works. Next, the method is applied to the optimization of a simple truss structure to explain how the method can be used in structural optimization. The same problems have been solved by penalty method and compared. The results from both methods have been the same. However, because of the long step length and reduction in the number of variables, the speed of convergence has been higher in the presented method.