Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T03:13:10.613Z Has data issue: false hasContentIssue false
  • English
  • Français

Development of a Parallel Explicit Finite-Volume Euler Equation Solver using the Immersed Boundary Method with Hybrid MPI-CUDA Paradigm

Published online by Cambridge University Press:  11 October 2019

F. A. Kuo ,
C. H. Chiang ,
M. C. Lo  and
J. S. Wu Open the ORCID record for J. S. Wu [Opens in a new window]
F. A. Kuo
Affiliation:
Department of Mechanical EngineeringNational Chiao Tung UniversityHsinchu, Taiwan National Center for High-Performance ComputingNational Applied Research LaboratoriesHsinchu, Taiwan
C. H. Chiang
Affiliation:
Department of Mechanical and Automation EngineeringI-Shou UniversityKaohsiung, Taiwan
M. C. Lo
Affiliation:
Department of Mechanical and Aerospace EngineeringChung Cheng Institute of Technology, National Defense UniversityTaoyuan, Taiwan
J. S. Wu*
Affiliation:
Department of Mechanical EngineeringNational Chiao Tung UniversityHsinchu, Taiwan
*
*Corresponding author ([email protected])
Get access

Abstract

This study proposed the application of a novel immersed boundary method (IBM) for the treatment of irregular geometries using Cartesian computational grids for high speed compressible gas flows modelled using the unsteady Euler equations. Furthermore, the method is accelerated through the use of multiple Graphics Processing Units – specifically using Nvidia’s CUDA together with MPI - due to the computationally intensive nature associated with the numerical solution to multi-dimensional continuity equations. Due to the high degree of locality required for efficient multiple GPU computation, the Split Harten-Lax-van-Leer (SHLL) scheme is employed for vector splitting of fluxes across cell interfaces. NVIDIA visual profiler shows that our proposed method having a computational speed of 98.6 GFLOPS and 61% efficiency based on the Roofline analysis that provides the theoretical computing speed of reaching 160 GLOPS with an average 2.225 operations/byte. To demonstrate the validity of the method, results from several benchmark problems covering both subsonic and supersonic flow regimes are presented. Performance testing using 96 GPU devices demonstrates a speed up of 89 times that of a single GPU (i.e. 92% efficiency) for a benchmark problem employing 48 million cells. Discussions regarding communication overhead and parallel efficiency for varying problem sizes are also presented.

Keywords

Euler equationsFinite volume methodImmersed boundary methodGPU
Type
Research Article
Information
Journal of Mechanics , Volume 36 , Issue 1 , February 2020 , pp. 87 - 102
Copyright
Copyright © 2019 The Society of Theoretical and Applied Mechanics 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Kamil, A. and Yelick, K., “Hierarchical Computation in the SPMD Programming Model,” Languages and Compilers for Parallel Computing SE - 1, vol. 8664, pp. 3–19, (2014).Google Scholar
Fluent, A., “ANSYS Fluent 12.0 user’s guide,” Ansys Inc, vol. 15317, pp. 1–2498, (2009).Google Scholar
Rodio, J. J., Xiao, X., Hassan, H. A., and Rumsey, C. L., “NASA Trapezoidal Wing Simulation using Stress-ω and One- and Two-Equation Turbulence Models,” Proceedings of AIAA SciTech 52nd Aerospace Sciences Meeting, pp. 1–19, (2014).CrossRefGoogle Scholar
Jasak, H., Jemcov, A., and Tukovic, Z., “OpenFOAM : A C++ Library for Complex Physics Simulations,” Proceedings of International Workshop on Coupled Methods in Numerical Dynamics, pp. 1–20, (2007).Google Scholar
Phillips, E., Zhang, Y., Davis, R., and Owens, J., “Rapid aerodynamic performance prediction on a cluster of graphics processing units,” Proceedings of the 47th AIAA Aerospace Sciences Meeting, pp. 1–11, (2009).CrossRefGoogle Scholar
Scheidegger, C. E., Comba, J. L. D., and da Cunha, R. D., “Practical CFD Simulations on Programmable Graphics Hardware using SMAC+,” Computer Graphics Forum, vol. 24, no. 4, pp. 715–728, (Dec. 2005).CrossRefGoogle Scholar
Ma, Z. H., Wang, H., and Pu, S. H., “GPU computing of compressible flow problems by a meshless method with space-filling curves,” Journal of Computational Physics, vol. 263, pp. 113–135, (Apr. 2014).CrossRefGoogle Scholar
Shahbazi, K., Hesthaven, J. S., and Zhu, X., “Multidimensional hybrid Fourier continuation–WENO solvers for conservation laws,” Journal of Computational Physics, vol. 253, pp. 209–225, (2013).CrossRefGoogle Scholar
Tuttafesta, M., Colonna, G., and Pascazio, G., “Computing unsteady compressible flows using Roe’s flux-difference splitting scheme on GPUs,” Computer Physics Communications, vol. 184, no. 6, pp. 1497–1510, (2013).CrossRefGoogle Scholar
Niemeyer, K. E. and Sung, C. J., “Recent progress and challenges in exploiting graphics processors in computational fluid dynamics,” The Journal of Supercomputing, vol. 67, no. 2, pp. 528–564, (Feb. 2014).Google Scholar
Kuo, F. A., Smith, M. R., Hsieh, C. W., Chou, C. Y., and Wu, J. S., “GPU acceleration for general conservation equations and its application to several engineering problems,” Computers & Fluids, vol. 45, pp. 147–154, (2011).CrossRefGoogle Scholar
Su, C. C., Smith, M. R., Kuo, F. A., Wu, J. S., Hsieh, C. W., and Tseng, K. C., “Large-scale simulations on multiple Graphics Processing Units (GPUs) for the direct simulation Monte Carlo method,” Journal of Computational Physics, vol. 231, pp. 7932–7958, (2012).CrossRefGoogle Scholar
Lo, M. C., Su, C. C., Wu, J. S., and Kuo, F. A., “Development of parallel direct simulation Monte Carlo method using a cut-cell Cartesian grid on a single graphics processor,” Computers & Fluids, vol. 101. pp. 114–125, (2014).CrossRefGoogle Scholar
Schneider, E. E. and Robertson, B. E., “Cholla: A New Massively Parallel Hydrodynamics Code For Astrophysical Simulation,” The Astrophysical Journal Supplement Series, vol. 217, no. 2, p. 24, (Apr. 2015).Google Scholar
Miao, S., Zhang, X., Parchment, O. G., and Chen, X., “A fast GPU based bidiagonal solver for computational aeroacoustics,” Computer Methods in Applied Mechanics and Engineering, vol. 286, pp. 22–39, (2015).CrossRefGoogle Scholar
Zimmerman, B. J. and Wang, Z. J., “The efficient implementation of correction procedure via reconstruction with graphics processing unit computing,” Computers & Fluids, vol. 101, pp. 263–272, (Sep. 2014).CrossRefGoogle Scholar
Fu, L., Gao, Z., Xu, K., and Xu, F., “A multi-block viscous flow solver based on GPU parallel methodology,” Computers & Fluids, vol. 95, pp. 19–39, (May 2014).CrossRefGoogle Scholar
Shinn, A. F., Goodwin, M. A., and Vanka, S. P., “Immersed boundary computations of shear- and buoyancy-driven flows in complex enclosures,” International Journal of Heat and Mass Transfer, vol. 52, no. 17–18, pp. 4082–4089, (2009).CrossRefGoogle Scholar
Harten, A., Lax, P., and Leer, B., “On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws,” SIAM Review, vol. 25, no. 1, pp. 35–61, (Jan. 1983).CrossRefGoogle Scholar
Peskin, C. S., “Flow patterns around heart valves: A numerical method,” Journal of Computational Physics, vol. 10, pp. 252–271, (1972).CrossRefGoogle Scholar
Mittal, R. and Iaccarino, G., “Immersed Boundary Methods,” Annual Review of Fluid Mechanics, vol. 37. pp. 239–261, (2005).CrossRefGoogle Scholar
Clarke, D. K., Hassan, H. A., and Salas, M. D., “Euler calculations for multielement airfoils using Cartesian grids,” AIAA Journal, vol. 24, no. 3, pp. 353–358, (Mar. 1986).CrossRefGoogle Scholar
Tseng, Y. H. and Ferziger, J. H., “A ghost-cell immersed boundary method for flow in complex geometry,” Journal of Computational Physics, vol. 192, pp. 593–623, (2003).CrossRefGoogle Scholar
Kontzialis, K. and Ekaterinaris, J. A., “High order discontinuous Galerkin discretizations with a new limiting approach and positivity preservation for strong moving shocks,” Computers & Fluids , vol. 71, pp. 98–112, (2013).CrossRefGoogle Scholar
Liu, J., Zhao, N., and Hu, O., “The ghost cell method and its applications for inviscid compressible flow on adaptive tree cartesian grids,” Advances in Applied Mathematics and Mechanics, vol. 1, pp. 664–682, (2009).CrossRefGoogle Scholar
Ghias, R., Mittal, R., and Dong, H., “A sharp interface immersed boundary method for compressible viscous flows,” Journal of Computational Physics, vol. 225, pp. 528–553, (2007).CrossRefGoogle Scholar
Marshall, D. D. and Ruffin, S. M., “An Embedded Boundary Cartesian Grid Scheme for Viscous Flows using a New Viscous Wall Boundary Condition Treatment,” Proceedings of 42nd AIAA Aerospace Sciences Meeting and Exhibit, pp. 1–8, (2004).CrossRefGoogle Scholar
Colella, P., Graves, D. T., Keen, B. J., and Modiano, D., “A Cartesian grid embedded boundary method for hyperbolic conservation laws,” Journal of Computational Physics, vol. 211, pp. 347–366, (2006).CrossRefGoogle Scholar
Toro, E. F., Riemann solvers and numerical methods for fluid dynamics-A Practical Introduction, (2009).Google Scholar
Davis, S. F., “Simplified Second-Order Godunov-Type Methods,” SIAM Journal on Scientific and Statistical Computing, vol. 9, no. 3, pp. 445–473, (May 1988).CrossRefGoogle Scholar
Burden, R. L. and Faires, J. D., Numerical analysis. Brooks/Cole, (2001).Google Scholar
Williams, S. W., “The Roofline Model,” Performance Tuning of Scientific Applications, pp. 195–215, (2011).CrossRefGoogle Scholar
Wermelinger, F., Hejazialhosseini, B., Hadjidoukas, P., Rossinelli, D., and Koumoutsakos, P., “An Efficient Compressible Multicomponent Flow Solver for Heterogeneous CPU/GPU Architectures,” Proceedings of the Platform for Advanced Scientific Computing Conference on ZZZ - PASC ’16, (2016).CrossRefGoogle Scholar
Williams, S., Waterman, A., and Patterson, D., “Roofline: an insightful visual performance model for multicore architectures,” Communications of the ACM, vol. 52, pp. 65–76, (2009).CrossRefGoogle Scholar
Demirdžić, I., Lilek, Ž., and Perić, M., “A collocated finite volume method for predicting flows at all speeds,” International Journal for Numerical Methods in fluids, vol. 16, no. January, pp. 1029–1050, (1993).CrossRefGoogle Scholar
Furmánek, P., “Numerical Solution of Steady and Unsteady Compressible Flow,” Czech Technical University in Prague, (2008).Google Scholar
Chang, S. C., Wang, X. Y., and Chow, C. Y., “The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws,” Journal of Computational Physics, vol. 136, pp. 89–136, (1998).Google Scholar
Sun, M. and Takayama, K., “Vorticity production in shock diffraction,” Journal of Fluid Mechanics, vol. 478, pp. 237–256, (2003).CrossRefGoogle Scholar