Improved gradient projection method for parametric optimisation of bar structures
Numerical optimization and the finite element method have been developed together to make possible the emergence of structural optimization as a potential design tool. The main research goal of this paper is the development of mathematical support and a numerical algorithm to solve parametric optimization problems of structures with orientation on software implementation in a computer-aided design system. The paper considers parametric optimization problems for bar structures formulated as nonlinear programming ones. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the parametric optimization problem. Equivalent Householder transformations of the resolving equations of the method have been proposed. They increase numerical efficiency of the algorithm developed based on the considered method. Additionally, proposed improvement for the gradient projection method also consists of equivalent Givens transformations of the resolving equations. They ensure acceleration of the iterative searching process in the specified cases described by the paper due to decreasing the amount of calculations. The comparison of the optimization results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient projection method. The efficiency of the proposed improvement of the gradient projection method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.