An operator splitting algorithm for Tikhonov-regularized topology optimization

In this work, we investigate a Tikhonov-type regularization scheme to address the ill-posedness of the classical compliance minimization problem. We observe that a semi-implicit discretization of the gradient descent flow for minimization of the regularized objective function leads to a convolution of the original gradient descent step with the Green's function associated with the modified Helmholtz equation. The appearance of " filtering" in this update scheme is different from the current density and sensitivity filtering techniques in the literature. The next iterate is defined as the projection of this provisional density onto the space of admissible density functions. For a particular choice of projection mapping, we show that the algorithm is identical to the well-known forward-backward splitting algorithm, an insight that can be further explored for topology optimization. Also of interest is that with an appropriate choice of the projection parameter, nearly all intermediate densities are eliminated in the optimal solution using the common density material models. We show examples of near binary solutions even for large values of the regularization parameter.