Abstract
This paper presents the general purpose framework Peano for the solution of partial differential equations (PDE) on adaptive Cartesian grids. The strict structuredness and inherent multilevel property of these grids allows for very low memory requirements, efficient (in terms of hardware performance) implementations of parallel multigrid solvers on dynamically adaptive grids, and arbitrary spatial dimensions. This combination of advantages distinguishes Peano from other PDE frameworks. We describe shortly the underlying octree-like grid type and its most important properties. The main part of the paper shows the framework concept of Peano and the implementation of a Navier–Stokes solver as one of the main currently implemented application examples. Various results ranging from hardware and numerical performance to concrete application scenarios close the contribution.
Similar content being viewed by others
References
Aoki T (1997) Interpolated differential operator (IDO) scheme for solving partial differential equations. Comput Phys Comm 102: 132–146
Bader M, Bungartz H-J, Frank A, Mundani R-P (2002) Space tree structures for PDE software. In: Proceedings of the International Conference on Computer Science (3), vol 2331, p 662
Bader M, Frank A, Zenger C (2002) An octree-based approach for fast elliptic solvers. High Perform Sci Eng Comput 21: 157–166
Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Ohlberger M, Sander O (2008) A generic grid interface for parallel and adaptive scientific computing. Part I: Abstract Framework. Computing 82(2–3): 103–119
Borrmann A, Schraufstetter S, Rank E (2009) Implementing metric operators of a spatial query language for 3d building models: octree and b-rep approaches. J Comput Civil Eng 23(1): 34–46
Brenk M, Bungartz H-J, Daubner K, Mehl M, Muntean IL, Neckel T (2008) An Eulerian approach for partitioned fluid-structure simulations on Cartesian grids. Comput Mech (accepted)
Brenk M, Bungartz H-J, Mehl M, Muntean IL, Neckel T, Weinzierl T (2008) Numerical simulation of particle transport in a drift ratchet. SIAM J Sci Comput 30(6): 2777–2798
Deering M (1995) Geometry compression. In: SIGGRAPH ’95: proceedings of the 22nd annual conference on computer graphics and interactive techniques. ACM Press, New York, pp 13–20
Düster A, Bröker H, Heidkamp H, Heißerer U, Kollmannsberger S, Krause R, Muthler A, Niggl A, Nübel V, Rücker M, Scholz D (2004) AdhoC4—user’s guide. Lehrstuhl für Bauinformatik, Technische Universität München
Fuster D, Baguéa A, Boeckc T, Le Moynea L, Leboissetierd A, Popinete S, Raya P, Scardovellif R, Zaleskia S (2009) Simulation of primary atomization with an octree adaptive mesh refinement and vof method. Int J Multiph Flow 35(6): 550–565
Gamma E, Helm R, Johnson RE, Vlissides J (1994) Design patterns—elements of reusable object-oriented software, 1st edn. Addison-Wesley, Longman
Gao F, Ingram DM, Causon DM, Mingham CG (2007) The development of a Cartesian cut cell method for incompressible viscous flows. Int J Numer Meth Fluids 64(9): 1033–1053
Gerstenbrger A, Wall WA (2007) An extended finite element method/mortar method based approach for fluid-structure interactions. Comput Methods Appl Mech Eng 197: 1699–1714
Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys Fluids 8(12): 2182–2189
Imai Y, Aoki T (2006) A higher-order implicit IDO scheme and its CFD application to local mesh refinement method. Comput Mech 38: 211–221
Imai Y, Aoki T, Takizawa K (2008) Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics. J Comput Phys 227: 2263–2285
Klass O, Shephard MS (2000) Automatic generation of octree-based three-dimensional discretisations for partition of unity methods. J Comput Mech 25(2–3): 296–304
Lam TW, Yu KM, Cheung KM, Li CL (1998) Octree reinforced thin shell objects rapid prototyping by fused deposition modelling. Int J Adv Manufact Technol 14(9): 631–636
Long K (2009) Sundance: a rapid prototyping toolkit for parallel pde simulation and optimization. In: Heinkenschloss M, Biegler LT, Ghattas O, Bloemen Waanders B (eds) Large-scale PDE-constrained optimization. Lecture notes in computational science and engineering, vol 30. Springer, Berlin, pp 331–339
Matthias S, Müller F (2003) Asymmetric pores in a silicon membrane acting as massively parallel brownian ratchets. Lett Nat 424: 53–57
Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37: 239–261
Morton GM (1966) A computer oriented geodetic data base and a new technique in file sequencing. Technical report, IBM Ltd., Ottawa, Ontario
Neckel T (2009) The PDE framework Peano: an environment for efficient flow simulations. Verlag Dr. Hut, München
Sagan H (1994) Space-filling curves. Springer, New York
Samet H (1984) The quadtree and related hierarchical data structures. ACM Comput Surv 16(2): 187–260
Sampath RS, Adavani SS, Sundar H, Lashuk I, Biros G (2008) Dendro: parallel algorithms for multigrid and amr methods on 2:1 balanced octrees. In: SC ’08: proceedings of the 2008 ACM/IEEE conference on supercomputing, Piscataway, NJ, USA. IEEE Press, New York, pp 1–12
Sundar H, Sampath RS, Biros G (2008) Bottom-up construction and 2:1 balance refinement of linear octrees in parallel. SIAM J Sci Comput 30(5): 2675–2708
Tomé MF, McKee S (1994) GENSMAC: a computational marker and cell method for free surface flows in general domains. J Comput Phys 110: 171–186
Tu T, O’Hallaron DR, Ghattas O (2005) Scalable parallel octree meshing for terascale applications. In: SC ’05: proceedings of the 2005 ACM/IEEE conference on supercomputing, Washington, DC, USA. IEEE Computer Society, New York, p 4
Turek S, Schäfer M (1996) Benchmark computations of laminar flow around a cylinder. In: Hirschel EH (ed) Flow simulation with high-performance computers II, NNFM, vol 52. Vieweg, Braunschweig
Vandevoorde D, Josuttis N (2003) C++ templates—the complete guide. Addison-Wesley, Reading
Weinzierl T (2009) A framework for parallel PDE solvers on multiscale adaptive Cartesian grids. Verlag Dr. Hut, München
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bungartz, HJ., Mehl, M., Neckel, T. et al. The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids. Comput Mech 46, 103–114 (2010). https://doi.org/10.1007/s00466-009-0436-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-009-0436-x