A new iterative method based on the Q-iteration method for robust and rapid determination of concentration distribution in multicomponent separation cascades

Abstract

Determining the concentration distribution of mixtures of multiple isotopes in separation cascades is a common task in cascade analysis and design. The robustness and efficiency of the computational method are crucial, particularly for cascade optimization. The Q-iteration method is well known for its simplicity, robustness, and rapid convergence and applies to various cascades. However, extensive experience in computation with the Q-iteration method revealed three drawbacks in certain cases: difficulties in selecting a suitable relaxation factor and appropriate update direction, as well as insufficient convergence speed. This paper analyzes the convergence properties of the Q-iteration method and its associated drawbacks and, based on the Q-iteration method, proposes a novel iterative approach, termed Minimal Residuals Q-iteration method (MR Q-iteration method), utilizing the gradient information of $Q$ to determine the update direction and step size at a low computational cost. Three typical cascade cases, each representing one of the drawbacks, are computed to compare the performances of both the Q-iteration method and the MR Q-iteration method. The results demonstrate that the new method effectively resolves the three drawbacks and has improved robustness and faster convergence compared to the Q-iteration method, while retaining its simplicity.