An improved gradient-based method to solve parametric optimisation problems of the bar structures

The paper considers parametric optimisation problems for the bar structures formulated as non-linear programming tasks. In the paper a gradient-based method is considered as investigated object. The main research question is the development of mathematical support and numerical algorithm to solve parametric optimisation problems of the building structures with orientation on software implementation in a computer-aided design system. 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 optimisation problem. Equivalent Householder transformations of the resolving equations of the method have been proposed by the paper. They increase numerical efficiency of the algorithm developed based on the method under consideration. Additionally, proposed improvement for the gradient-based 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. Lengths of the gradient vectors for objective function, as well as for constraints remain as they were in scope of the proposed equivalent transformations ensuring the reliability of the optimisation algorithm. The comparison of the optimisation results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient-based method. Start values of the design variables have no influence on the optimum solution of the non-linear problem confirming in such way accuracy and validity of the optimum solutions obtained using the algorithm developed based on the presented improved gradient-based method. The efficiency of the propoced improvement of the gradient-based method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.