Acceleration of a BEM based solution of the velocity-vorticity formulation of the Navier-Stokes equations by the cross approximation method

摘要

In this paper, we present a method to decrease the computational demand of the boundary-domain integral equation based 3D flow solver. We focus on the solution of the velocity-vorticity formulation of the Navier-Stokes equation, which governs incompressible viscous fluid flow. We introduce the cross approximation into the solution of the boundary vorticity values. This problem is governed by the Poisson type kinematics equation and presents a computational bottleneck of the algorithm. In order to accelerate the flow solver, we approximate the domain contribution of the kinematics integral equation by the cross approximation algorithm. The cross approximation method is used in combination with the hierarchical decomposition of the domain boundary combined by the hierarchical decomposition of domain interior. We propose to specify the approximation extent by controlling the depth of the hierarchical decomposition and the rank of the approximated integral matrix parts. The developed algorithm is tested using the Arnold-Beltrami-Childress and lid driven cavity flows. We study the accuracy of boundary vorticity estimation and of the flow solution for different flow complexities (Reynolds number values), computational mesh densities and cross approximation settings. We found that that by using the cross approximation technique in the flow solver, we were able to reduce the computational demands of storing matrices to approximately 30% of the storage space of the original matrices. Furthermore, we showed that achieved approximation extent depends on the complexity of the simulated flow problem.