zbMATH — the first resource for mathematics

The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids. (English) Zbl 1301.76056
Summary: 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.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65Y15 Packaged methods for numerical algorithms
Full Text: DOI
[1] Aoki T (1997) Interpolated differential operator (IDO) scheme for solving partial differential equations. Comput Phys Comm 102: 132–146 · doi:10.1016/S0010-4655(97)00020-9
[2] 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 · Zbl 1056.65107
[3] Bader M, Frank A, Zenger C (2002) An octree-based approach for fast elliptic solvers. High Perform Sci Eng Comput 21: 157–166
[4] 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 · Zbl 1151.65089 · doi:10.1007/s00607-008-0003-x
[5] 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 · doi:10.1061/(ASCE)0887-3801(2009)23:1(34)
[6] 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) · Zbl 1228.74024
[7] 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 · Zbl 1185.35159 · doi:10.1137/070692212
[8] 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
[9] 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
[10] 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 · doi:10.1016/j.ijmultiphaseflow.2009.02.014
[11] Gamma E, Helm R, Johnson RE, Vlissides J (1994) Design patterns–elements of reusable object-oriented software, 1st edn. Addison-Wesley, Longman
[12] 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 · Zbl 1178.76275 · doi:10.1002/fld.1409
[13] 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 · Zbl 1194.76117 · doi:10.1016/j.cma.2007.07.002
[14] 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 · Zbl 1180.76043 · doi:10.1063/1.1761178
[15] 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 · Zbl 1176.76085 · doi:10.1007/s00466-005-0742-x
[16] 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 · Zbl 1261.76031
[17] 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 · Zbl 0956.65113 · doi:10.1007/s004660050478
[18] 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 · doi:10.1007/BF01192282
[19] 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
[20] Matthias S, Müller F (2003) Asymmetric pores in a silicon membrane acting as massively parallel brownian ratchets. Lett Nat 424: 53–57 · doi:10.1038/nature01736
[21] Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37: 239–261 · Zbl 1117.76049 · doi:10.1146/annurev.fluid.37.061903.175743
[22] Morton GM (1966) A computer oriented geodetic data base and a new technique in file sequencing. Technical report, IBM Ltd., Ottawa, Ontario
[23] Neckel T (2009) The PDE framework Peano: an environment for efficient flow simulations. Verlag Dr. Hut, München
[24] Sagan H (1994) Space-filling curves. Springer, New York · Zbl 0806.01019
[25] Samet H (1984) The quadtree and related hierarchical data structures. ACM Comput Surv 16(2): 187–260 · doi:10.1145/356924.356930
[26] 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
[27] 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 · Zbl 1186.68554 · doi:10.1137/070681727
[28] 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 · Zbl 0790.76058 · doi:10.1006/jcph.1994.1013
[29] 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
[30] 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
[31] Vandevoorde D, Josuttis N (2003) C++ templates–the complete guide. Addison-Wesley, Reading
[32] Weinzierl T (2009) A framework for parallel PDE solvers on multiscale adaptive Cartesian grids. Verlag Dr. Hut, München
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.