×

Topology optimization of unsteady flows using the spectral element method. (English) Zbl 1521.76618

Summary: We investigate the applicability of a high-order Spectral Element Method (SEM) to density based topology optimization of unsteady flows in two dimensions. Direct Numerical Simulations (DNS) are conducted relying on Brinkman penalization to describe the presence of solid within the domain. The optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. A nonlinear filtering strategy is used to both enforce a minimum length scale and to provide smoothing across the fluid-solid interface, preventing Gibbs oscillations. This method has been successfully applied to the design of a channel bend and an oscillating pump, and demonstrates good agreement with body fitted meshes. The precise design of the pump is shown to depend on the initial material distribution. However, the underlying topology and pumping mechanism is the same. The effect of a minimum length scale has been studied, revealing it to be a necessary regularization constraint for the oscillating pump to produce meaningful designs. The combination of SEM and density based optimization offer some unique challenges which are addressed and discussed, namely a lack of explicit boundary tracking exacerbated by the interface smoothing. Nevertheless, SEM can achieve equivalent levels of precision to traditional finite element methods, while requiring fewer degrees of freedom. Hence, the use of SEM addresses the two major bottlenecks associated with optimizing unsteady flows: computation cost and data storage.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76F65 Direct numerical and large eddy simulation of turbulence
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids

Software:

Nek5000; revolve
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bendsøe, Martin Philip; Kikuchi, Noboru, Generating optimal topologies in structural design using a homogenization method, Comput Methods Appl Mech Eng, 71, 2 (1988) · Zbl 0671.73065
[2] Borrvall, Thomas; Petersson, Joakim, Topology optimization of fluids in Stokes flow, Int J Numer Methods Fluids, 41, 1 (2003) · Zbl 1025.76007
[3] Gersborg-Hansen, A.; Sigmund, O.; Haber, R. B., Topology optimization of channel flow problems, Struct Multidiscip Optim, 30, 3 (2005) · Zbl 1243.76034
[4] Kreissl, Sebastian; Pingen, Georg; Maute, Kurt, Topology optimization for unsteady flow, Int J Numer Methods Eng, 87, 13 (2011) · Zbl 1242.76052
[5] Pingen, Georg; Maute, Kurt, Optimal design for non-Newtonian flows using a topology optimization approach, Comput Math Appl, 59, 7 (2010) · Zbl 1193.76113
[6] Saglietti Clio, Wadbro Eddie, Berggren Martin, Henningson Dan S. Heat transfer maximization in a three dimensional conductive differentially heated cavity by means of topology optimization. In: Proceedings of the 6th european conference on computational mechanics: solids, structures and coupled problems, ECCM 2018 and 7th european conference on computational fluid dynamics, ECFD 2018; 2020.
[7] Saglietti, Clio; Schlatter, Philipp; Wadbro, Eddie; Berggren, Martin; Henningson, Dan S., Topology optimization of heat sinks in a square differentially heated cavity, Int J Heat Fluid Flow, 74 (2018)
[8] Alexandersen, Joe; Andreasen, Casper Schousboe, A review of topology optimisation for fluid-based problems, Fluids, 5, 1 (2020)
[9] CHEN, Cong; YAJI, Kentaro; YAMADA, Takayuki; IZUI, Kazuhiro; NISHIWAKI, Shinji, Local-in-time adjoint-based topology optimization of unsteady fluid flows using the lattice Boltzmann method, Mech Eng J, 4, 3 (2017)
[10] Griewank, Andreas; Walther, Andrea, Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans Math Softw, 26, 1 (2000) · Zbl 1137.65330
[11] Challis, Vivien J.; Guest, James K., Level set topology optimization of fluids in Stokes flow, Int J Numer Methods Eng, 79, 10 (2009) · Zbl 1176.76039
[12] Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, Int J Numer Methods Fluids, 58, 8 (2008) · Zbl 1152.76025
[13] Koch, J. R. L.; Papoutsis-Kiachagias, E. M.; Giannakoglou, K. C., Transition from adjoint level set topology to shape optimization for 2D fluid mechanics, Comput Fluids, 150 (2017) · Zbl 1390.76057
[14] Nørgaard, Sebastian Arlund, Topology optimization and lattice Boltzmann methods, DCAMM Special Report, S230 (2017), Technical University of Denmark, (Ph.D. thesis)
[15] Nørgaard, Sebastian; Sigmund, Ole; Lazarov, Boyan, Topology optimization of unsteady flow problems using the lattice Boltzmann method, J Comput Phys, 307 (2016) · Zbl 1351.76245
[16] Regulski, W.; Szumbarski, J.; Łaniewski-Wołłk, Boyan; Gumowski, K.; Skibiński, J.; Wichrowski, M.; Wejrzanowski, T., Pressure drop in flow across ceramic foams-A numerical and experimental study, Chem Eng Sci, 137 (2015)
[17] Yu, Dazhi; Mei, Renwei; Luo, Li Shi; Shyy, Wei, Viscous flow computations with the method of lattice Boltzmann equation, Prog Aerosp Sci, 39, 5 (2003)
[18] Kontoleontos, E. A.; Papoutsis-Kiachagias, E. M.; Zymaris, A. S.; Papadimitriou, D. I.; Giannakoglou, K. C., Adjoint-based constrained topology optimization for viscous flows, including heat transfer, Eng Optim, 45, 8 (2013)
[19] Dilgen, Cetin B.; Dilgen, Sumer B.; Fuhrman, David R.; Sigmund, Ole; Lazarov, Boyan S., Topology optimization of turbulent flows, Comput Methods Appl Mech Eng, 331 (2018) · Zbl 1439.74265
[20] Kreissl, Sebastian; Maute, Kurt, Levelset based fluid topology optimization using the extended finite element method, Struct Multidiscip Optim, 46, 3 (2012) · Zbl 1274.76251
[21] Villanueva, Carlos H.; Maute, Kurt, CutFEM topology optimization of 3D laminar incompressible flow problems, Comput Methods Appl Mech Eng, 320 (2017) · Zbl 1439.74297
[22] Duan, Xian Bao; Li, Fei Fei; Qin, Xin Qiang, Adaptive mesh method for topology optimization of fluid flow, Appl Math Lett, 44 (2015) · Zbl 1311.76021
[23] Feppon, F.; Allaire, G.; Dapogny, C.; Jolivet, P., Topology optimization of thermal fluid-structure systems using body-fitted meshes and parallel computing, J Comput Phys, 417 (2020) · Zbl 1437.74021
[24] Wang, Z. J.; Fidkowski, Krzysztof; Abgrall, Rémi; Bassi, Francesco; Caraeni, Doru; Cary, Andrew; Deconinck, Herman; Hartmann, Ralf; Hillewaert, Koen; Huynh, H. T.; Kroll, Norbert; May, Georg; Persson, Per Olof; van Leer, Bram; Visbal, Miguel, High-order CFD methods: Current status and perspective, Int J Numer Methods Fluids, 72, 8 (2013) · Zbl 1455.76007
[25] Fischer, Paul; Mullen, Julia, Filter-based stabilization of spectral element methods, Comptes Rendus de l’Académie des Sci - Ser I - Math, 332, 3 (2001) · Zbl 0990.76064
[26] Hosseini, S. M.; Vinuesa, R.; Schlatter, P.; Hanifi, A.; Henningson, D. S., Direct numerical simulation of the flow around a wing section at moderate Reynolds number, Int J Heat Fluid Flow, 61 (2016)
[27] Offermans, Nicolas; Marin, Oana; Schanen, Michel; Gong, Jing; Fischer, Paul; Schlatter, Philipp, On the strong scaling of the spectral element solver Nek5000 on petascale systems, arxiv (2017)
[28] Vincent, Jonathan; Gong, Jing; Karp, Martin; Peplinski, Adam; Jansson, Niclas; Podobas, Artur; Jocksch, Andreas; Yao, Jie; Hussain, Fazle; Markidis, Stefano; Karlsson, Matts; Pleiter, Dirk; Laure, Erwin; Schlatter, Philipp, Strong scaling of OpenACC enabled Nek5000 on several GPU based HPC systems (2021)
[29] Ghasemi, Ali; Elham, Ali, A novel topology optimization approach for flow power loss minimization across fin arrays, Energies, 13, 8 (2020)
[30] Wadbro, Eddie; Hägg, Linus, On quasi-arithmetic mean based filters and their fast evaluation for large-scale topology optimization, Struct Multidiscip Optim, 52, 5 (2015)
[31] Bendsøe, M. P.; Sigmund, O., Topology optimization using the finite volume method, Topology, m, June (2005)
[32] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip flow boundary with an external force field, J Comput Phys, 105, 2 (1993) · Zbl 0768.76049
[33] Angot, Philippe; Bruneau, Charles Henri; Fabrie, Pierre, A penalization method to take into account obstacles in incompressible viscous flows, Numerische Mathematik, 81, 4 (1999) · Zbl 0921.76168
[34] Sigmund, Ole, Morphology-based black and white filters for topology optimization, Struct Multidiscip Optim, 33, 4-5 (2007)
[35] Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct Optim, 16, 1 (1998)
[36] Svanberg, Krister; Svärd, Henrik, Density filters for topology optimization based on the pythagorean means, Struct Multidiscip Optim, 48, 5 (2013)
[37] Guest, J. K.; Prévost, J. H.; Belytschko, T., Achieving minimum length scale in topology optimization using nodal design variables and projection functions, Int J Numer Methods Eng, 61, 2 (2004) · Zbl 1079.74599
[38] Wang, Fengwen; Lazarov, Boyan Stefanov; Sigmund, Ole, On projection methods, convergence and robust formulations in topology optimization, Struct Multidiscip Optim, 43, 6 (2011) · Zbl 1274.74409
[39] Xu, Shengli; Cai, Yuanwu; Cheng, Gengdong, Volume preserving nonlinear density filter based on heaviside functions, Struct Multidiscip Optim, 41, 4 (2010) · Zbl 1274.74419
[40] Hägg, Linus; Wadbro, Eddie, On minimum length scale control in density based topology optimization, Struct Multidiscip Optim, 58, 3 (2018)
[41] Deville, M. O.; Fischer, P. F.; Mund, E. H., (High-Order Methods for Incompressible Fluid Flow (2002)) · Zbl 1007.76001
[42] Maday Yvon, Patera Anthony T. Spectral element methods for the incompressible Navier-Stokes equations. In: State-of-the-Art surveys on computational mechanics. · Zbl 1012.65109
[43] Ohlsson, J.; Schlatter, P.; Fischer, P. F.; Henningson, D. S., Stabilization of the spectral-element method in turbulent flow simulations, (Lecture Notes in Computational Science and Engineering, vol. 76 LNCSE (2011)) · Zbl 1430.76315
[44] Malm, Johan; Schlatter, Philipp; Fischer, Paul F.; Henningson, Dan S., Stabilization of the spectral element method in convection dominated flows by recovery of skew-symmetry, J Sci Comput, 57, 2 (2013) · Zbl 1282.76145
[45] Schanen, Michel; Marin, Oana; Zhang, Hong; Anitescu, Mihai, Asynchronous two-level checkpointing scheme for large-scale adjoints in the spectral-element solver Nek5000, (Procedia Computer Science, vol. 80 (2016))
[46] Rinaldi, Enrico; Canton, Jacopo; Schlatter, Philipp, The vanishing of strong turbulent fronts in bent pipes, J Fluid Mech, 866 (2019) · Zbl 1415.76284
[47] Offermans, Nicolas, Aspects of adaptive mesh refinement in the spectral element method, TRITA-MEK, 2019:28, 206 (2019), KTH, Mechanics, (Ph.D. thesis)
[48] Svanberg, Krister, MMA and GCMMA - Two methods for nonlinear optimization, kth 1 (2007) · Zbl 1273.74407
[49] Deng, Yongbo; Wu, Yihui; Liu, Zhenyu, (Topology Optimization Theory for Laminar Flow (2018)) · Zbl 1386.76004
[50] Villanueva, Carlos H.; Maute, Kurt, CutFEM topology optimization of 3D laminar incompressible flow problems, Comput Methods Appl Mech Eng, 320, 444-473 (2017) · Zbl 1439.74297
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.