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[1] Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press: Cambridge, 1992.
[2] Greengard, L. & Rokhlin, V., The Rapid Evaluation of Potential Fields in Three Dimen-sions, In Vortex Methods, Springer Verlag, pp. 121–141, 1988.
[3] Greengard, L., The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, 1988.
[4] Hackbusch, W., A Sparse Matrix Arithmetic Based on H-Matrices. Part I. Introduction to H-matrices, Computing, 62(2), pp. 89–108, 1999.
[5] Hackbusch, W., Hierarchische Matrizen, Springer, 2009.
[6] Alouges, F. & Aussal, M., The sparse cardinal sine decomposition and its application for fast numerical convolution. Numerical Algorithms, 70(2), pp. 427–448, 2015. [Crossref]
[7] Dutt, A. & Rokhlin, V., Fast fourier transforms for nonequispaced data. SIAM Journal on Scientific Computing, 14(6), pp. 1368–1393, 1993. [Crossref]
[8] Lee, J.-Y. & Greengard, L., The type 3 nonuniform FFT and its application. Journal of Computational Physics, 206(1), pp. 1–5, 2005. [Crossref]
[9] Alouges, F., Aussal, M., Lefebvre-Lepot, A., Pigeonneau, F. & Sellier, A. The sparse cardinal sine decomposition applied to Stokes integral equations. Proceedings of the ICMF-2016, 9th International Conference on Multiphase Flow, Florence, 2016.
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Open Access
Research article

Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows

f. alouges1,
m. aussal1,
a. lefebvre-lepot1,
f. pigeonneau2,
a. sellier3
1
Centre de Mathématiques Appliquées, Ecole polytechnique, France
2
Surface du Verre et Interfaces, Saint-Gobain, France
3
LadHyX, Ecole polytechnique, France
International Journal of Computational Methods and Experimental Measurements
|
Volume 5, Issue 3, 2017
|
Pages 387-394
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: 03-31-2017
View Full Article|Download PDF

Abstract:

In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decompo- sition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence.

Keywords: boundary element method, fast convolution, Stokes equations
Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References
[1] Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press: Cambridge, 1992.
[2] Greengard, L. & Rokhlin, V., The Rapid Evaluation of Potential Fields in Three Dimen-sions, In Vortex Methods, Springer Verlag, pp. 121–141, 1988.
[3] Greengard, L., The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, 1988.
[4] Hackbusch, W., A Sparse Matrix Arithmetic Based on H-Matrices. Part I. Introduction to H-matrices, Computing, 62(2), pp. 89–108, 1999.
[5] Hackbusch, W., Hierarchische Matrizen, Springer, 2009.
[6] Alouges, F. & Aussal, M., The sparse cardinal sine decomposition and its application for fast numerical convolution. Numerical Algorithms, 70(2), pp. 427–448, 2015. [Crossref]
[7] Dutt, A. & Rokhlin, V., Fast fourier transforms for nonequispaced data. SIAM Journal on Scientific Computing, 14(6), pp. 1368–1393, 1993. [Crossref]
[8] Lee, J.-Y. & Greengard, L., The type 3 nonuniform FFT and its application. Journal of Computational Physics, 206(1), pp. 1–5, 2005. [Crossref]
[9] Alouges, F., Aussal, M., Lefebvre-Lepot, A., Pigeonneau, F. & Sellier, A. The sparse cardinal sine decomposition applied to Stokes integral equations. Proceedings of the ICMF-2016, 9th International Conference on Multiphase Flow, Florence, 2016.
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Alouges, F., Aussal, M., Lefebvre-lepot, A., Pigeonneau, F., & Sellier, A. (2017). Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows. Int. J. Comput. Methods Exp. Meas., 5(3), 387-394. https://doi.org/10.2495/CMEM-V5-N3-387-394
F. Alouges, M. Aussal, A. Lefebvre-lepot, F. Pigeonneau, and A. Sellier, "Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows," Int. J. Comput. Methods Exp. Meas., vol. 5, no. 3, pp. 387-394, 2017. https://doi.org/10.2495/CMEM-V5-N3-387-394
@research-article{Alouges2017ApplicationOT,
title={Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows},
author={F. Alouges and M. Aussal and A. Lefebvre-Lepot and F. Pigeonneau and A. Sellier},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2017},
page={387-394},
doi={https://doi.org/10.2495/CMEM-V5-N3-387-394}
}
F. Alouges, et al. "Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows." International Journal of Computational Methods and Experimental Measurements, v 5, pp 387-394. doi: https://doi.org/10.2495/CMEM-V5-N3-387-394
F. Alouges, M. Aussal, A. Lefebvre-Lepot, F. Pigeonneau and A. Sellier. "Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows." International Journal of Computational Methods and Experimental Measurements, 5, (2017): 387-394. doi: https://doi.org/10.2495/CMEM-V5-N3-387-394
ALOUGES F， AUSSAL M， LEFEBVRE-LEPOT A， et al. Application of the Sparse Cardinal Sine decomposition to 3D Stokes Flows[J]. International Journal of Computational Methods and Experimental Measurements， 2017， 5(3)： 387-394. https://doi.org/10.2495/CMEM-V5-N3-387-394