A very simple, efficient and accurate dynamic interpolation BFGS algorithm for three-dimensional moment of fluid method

Abstract

The Moment of Fluid (MOF) method is a widely used and accurate interface reconstruction method. It performs iterations to solve an implicit nonlinear optimization problem to obtain the optimal approximate interface. As far as we know, the current optimization algorithms employed in MOF methods all directly minimize the implicit objective function, and the interface reconstruction procedure has to be executed for each calculation of the objective function, which is very expensive. There have been numerous measures to improve the computational efficiency, accuracy and robustness of the MOF method, but we will achieve the improvements from a new perspective. In this paper, a very simple, efficient and accurate dynamic interpolation BFGS (DIBFGS) algorithm is proposed, based on the classical Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The new algorithm introduces a dynamic interpolation function to approximate the implicit objective function during the optimization process. The dynamic interpolation function is explicit and the interpolation nodes utilized for constructing it are dynamically selected from a gradually expanding set of candidate interpolation nodes. The new algorithm minimizes the dynamic interpolation function instead of the original objective function, thus the number of original objective function calculations will be significantly reduced and the computational efficiency can be improved. Moreover, in previous MOF methods, the gradient of the objective function involved in the iterations are either estimated by finite difference approximation or derived analytically with complex analyses, both of which require interface reconstruction. On the contrary, our algorithm uses the analytical gradient of the interpolation function, which is also explicit and quite easy to calculate. Then the accuracy, convergence rate as well as robustness of the iteration process can be improved. Last but not least, our algorithm is very simple and easy to code. A variety of numerical tests collectively show that compared to the MOF method using BFGS algorithm, our new method can not only effectively reduce the number of original objective function calculations and computational time, but also improve the accuracy and robustness particularly in complex cases with large interface deformations.