Acceleration of BEM with the Cross Approximation for Determination of Boundary Vorticity

In this paper, we present a fast boundary element method (BEM) algorithm for the solution of the velocity-vorticity formulation of the Navier-Stokes equations. The Navier-Stokes equations govern incompressible fluid flow, which is inherently nonlinear and when discretizised by BEM requires the discretization of the domain and calculation of domain integrals. The computational demands of such method scale with O(N2), where N is the number of boundary nodes. To accelerate the solution process and reduce the computational demand, we present two different approaches, the subdomain method and an approximation procedure with hierarchical structure. Several approximation techniques exist, such as multipole approximation methods FMM (fast multiple method), SVD (singular value decomposition method), wavelet transform method and a cross approximation method. In this paper, we present the cross approximation method in combination with the hierarchical H-structure. The cross approximation method can reduce the computational demands from O(N2) to O(N log N). There are many forms of the cross approximation, like the algebraic cross approximation and the hybrid cross approximation. Here, we applied the algebraic cross approximation form. The main advantage is that we did not need to evaluate the integral and then to change it with a degenerate kernel function. The cross approximation algorithm was used to solve the kinematics equation for unknown boundary vorticity values. Results show that an increasing of the compression rate has a negative influence on the solution accuracy. On the other hand, the solution accuracy increases with computational grid density. Tests were performed using the 3D lid-driven cavity test case with Reynolds numbers up to 1000. Solution accuracy was similar for all Reynolds numbers considered. In conclusion, the tests showed that our implementation of the algebraic cross approximation for the acceleration of the solution of the kinematics equation can be applied to decrease the computational demands and to accelerate the BEM.