Direct gradient projection method with transformation of variables technique for structural topology optimization

This paper proposes an efficient and reliable topology optimization method that can obtain a black and white solution with a low objective function value within a few tens of iterations. First of all, a transformation of variables technique is adopted to eliminate the constraints on the design variables. After that, the optimization problem is considered as aiming at the minimum compliance in the space of design variables which is supposed to be solved by iterative method. Based on the idea of the original gradient projection method, the direct gradient projection method (DGP) is proposed. By projecting the negative gradient of objective function directly onto the hypersurface of the constraint, the most promising search direction from the current position is obtained in the vector space spanned by the gradients of objective and constraint functions. In order to get a balance between efficiency and reliability, the step size is constrained in a rational range via a scheme for step size modification. Moreover, a grey elements suppression technique is proposed to lead the optimization to a black and white solution at the end of the process. Finally, the performance of the proposed method is demonstrated by three numerical examples including both 2D and 3D problems in comparison with the typical SIMP method using the optimality criteria algorithm.