OPEN ACCESS
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.
adaptive cross approximation, boundary element method, boundary-domain integral method, fast algorithms, hierarchical structure, H-matrix, kinematics equation, lid-driven cavity, velocity-vorticity formulation
[1] Wrobel, L.C., The boundary element method, John Willey and Sons, 2002.
[2] Börm, S., Grasedyck, L. & Hackbusch, W., Introduction to hierarchical matrices with applications. Engineering Analysis with Boundary Elements, 27(5), pp. 405–422, 2003. https://doi.org/10.1016/s0955-7997(02)00152-2
[3] Hackbusch, W., A sparse matrix arithmetic based on H -matrices. Computing, 108, pp. 89–108, 1999.https://doi.org/10.1007/s006070050015
[4] Börm, S., Approximation of integral operators by H2-matrices with adaptive bases. Computing, 74(3), pp. 249–271, 2005.https://doi.org/10.1007/s00607-004-0106-y
[5] Bebendorf, M., Approximation of boundary element matrices. Numerische Mathematik, 86(4), pp. 565–589, 2000.https://doi.org/10.1007/pl00005410
[6] Rjasanow, S. & Steinbach, O., The fast solution of boundary integral equations, Springer Science+Business Media, 2007.
[7] Tamayo, J.M., Heldring, A. & Rius, J.M., Multilevel adaptive cross approxima-tion (MLACA), IEEE Antennas and Propagation Society International Symposium Antennas and Propagation, pp. 2008–2011, 2009.
[8] Bebendorf, M. & Rjasanow, S., Adaptive low-rank approximation of collocation matrices. Computing, 70(1), pp. 1–24, 2003.https://doi.org/10.1007/s00607-002-1469-6
[9] Rjasanow, S., Adaptive cross approximation of dense matrices, IABEM 2002 Sympo-sium, pp. 1–12, 2002.
[10] Börm, S. & Grasedyck, L., Hybrid cross approximation of integral operators, Numerische Mathematik, 101(2), pp. 221–249, 2005.https://doi.org/10.1007/s00211-005-0618-1
[11] Schröder, A., Brüns, H.-D. & Schuster, C., Fast evaluation of electromagnetic fields using a parallelized adaptive cross approximation. Antennas and Propagation, IEEE Transactions on, 62(5), pp. 2818–2822, 2014.https://doi.org/10.1109/tap.2014.2303819
[12] Kurz, S., Rain, O. & Rjasanow, S., Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems. Computational Mechanics, 32(4), pp. 423–429, 2003.https://doi.org/10.1007/s00466-003-0511-7
[13] Grytsenko, T. & Galybin, A.N., Numerical analysis of multi-crack large-scale plane problems with adaptive cross approximation and hierarchical matrices. Engineering Analysis with Boundary Elements, 34(5), pp. 501–510, 2010.https://doi.org/10.1016/j.enganabound.2009.12.001
[14] Maerten, F., Adaptive cross-approximation applied to the solution of system of equa-tions and post-processing for 3D elastostatic problems using the boundary element method. Engineering Analysis with Boundary Elements, 34(5), pp. 483–491, 2010. https://doi.org/10.1016/j.enganabound.2009.10.016
[15] Wei, X., Chen, B., Chen, S. & Yin, S., An ACA-SBM for some 2D steady-state heat conduction problems. Engineering Analysis with Boundary Elements, 71, pp. 101–111, 2016.https://doi.org/10.1016/j.enganabound.2016.07.012
[16] Škerget, L., Hriberšek, M. & Žunic, Z., Natural convection flows in complex cavities by BEM. International Journal of Numerical Methods for Heat & Fluid Flow, 13(6), pp. 720–735, 2003.https://doi.org/10.1108/09615530310498394
[17] Ravnik, J., Škerget, L. & Zunič, Z., Velocity-vorticity formulation for 3D natural con-vection in an inclined enclosure by BEM. International Journal of Heat and Mass Transfer, 51(17–18), pp. 4517–4527, 2008.https://doi.org/10.1016/j.ijheatmasstransfer.2008.01.018
[18] Yang, J., Yang, S., Chen, Y. & Hsu, C., Implicit weighted ENO schemes for the three-dimensional incompressible Navier – Stokes equations. Journal of Computational Physics, 487, pp. 464–487, 1998.https://doi.org/10.1006/jcph.1998.6062